Learning probability in the Kingdom of Tonga: the influence of language and culture

Abstract

Problems teaching probability in Tonga (in the South Pacific) led to the question on how language and culture affect the understanding of probability and uncertainty. The research uses a discursive approach to identify the endorsed narratives which underlie Tongans’ reasoning in situations of uncertainty. I aim to justify the claim that the Tongan language and the Tongan way of life interact to make the concept of uncertainty very different from that found in western countries and the concept of probability almost redundant in Tongan day-to-day discourse. There are very few cross-cultural studies concerning the ways in which probability and uncertainty are understood in different cultural contexts, and this article aims to make a small contribution to filling this gap.

1 Introduction—the puzzle of learning probability in Tonga

Little has been written about the impact of culture and language on how probability is understood and how it should be taught. In this article, I present a case study based on research in the Kingdom of Tonga. It is to be hoped that the research outlined here will contribute to the small but growing literature concerning the effects of culture and language on mathematical subjects (Barton & Frank, 2001; Saxe, 2012; Presmeg, 2007; Meaney, Trinick, & Fairhall, 2012; and others). The field work described in this article was restricted to the island of Tongatapu (the largest island in Tonga), but the conclusions can be applied on a larger scale to the teaching of probability in other communities with different linguistic and cultural identities.

The research was motivated by unusual and puzzling difficulties which I experienced while trying to teach probability to college level students, who were otherwise proficient in mathematics. The seeds of the research were planted in a lesson devoted to basic ideas of uncertainty. The following conversation took place while I was attempting to teach how future events can be assigned probabilities:

Lecturer: What is the probability that the sun will be shining at this time tomorrow?

Student: It will be sunny.

Lecturer: Are you certain that it will be sunny?

Student: Yes.

Lecturer: Why are you certain?

Student: Because the angels told me.

This conversation led me to realise the futility of my attempts to evoke in students the kind of thinking on which the mathematical concept of probability could later be built. It eventually caused me to question my own, somewhat ethnocentric, definitions of uncertainty and probability. The respondent was an otherwise successful mathematics student and spoke good English. I was puzzled, and this motivated the study presented in this paper, an attempt to understand the way of thinking that lay behind my student’s response.

Preliminary observations indicated that the Tongan language does not provide Tongan students with the tools and the intuitive ideas which are so important in developing the notions of uncertainty and of probability. Although secondary and tertiary education was supposed to be in English, my students regularly switched to Tongan when discussing what I was teaching. Tongan is the language that mediates and organises these students’ lives and activities. It is an integral part of their culture and everyday life. More generally, the hypothesis to be tested is that the linguistic tools differ significantly from European languages and in particular the western concepts of probability and uncertainty do not exist in the community of native Tongan discourse. This said, as the traditional Tongan culture is being diluted, both the discourse on uncertainty and activities that require probabilistic thinking are changing.

As the research progressed, it became clear that language cannot be analysed on its own but that the culture and the history of Tonga have to be understood in order to explain the puzzling phenomenon which I had observed. Why did my students have a specific problem with probability, and not with other mathematical subjects? Can this fact be accounted for by particularities of the Tongan language and cultural practices?

As an outsider studying the Tongan culture, I was aware of the danger of judging the culture, beliefs, and values that I observed from my own “western” view point. As described in this article, Tonga has a long tradition of searching for a synthesis between the best values from the outside world while keeping the best parts of traditional culture. I was helped by my many Tongan friends and colleagues to respect and admire this approach.

The research was designed to answer two questions:

As a precondition of the research, it was necessary to test the claim that the discourse on probability poses specific difficulties to Tongan students in contrast to other subjects on the mathematics curriculum. Thus, the first research question:

• RQ1: Is it justified to claim that the discourse on probability is less accessible to Tongan students than other mathematical discourses (e.g., algebra, analysis, and geometry)?

If the answer to the first question is positive then the second research question follows:

• RQ2: What is the Tongan discourse about uncertainty and the probability of future events, and how can this account for the difficulties identified in the answer to RQ1?.

To answer the second question, it was necessary to look at both the Tongan language and cultural practices, specifically in relation to uncertainty and probabilistic thinking.

2 Theoretical background

2.1 A brief introduction to Tonga and the Tongan way (Anga Fakatonga)

Tonga is situated culturally and physically in an area of the Pacific Ocean known as Polynesia. Though there are great distances between the various island groups that constitute Polynesia, they have much in common. Before the Europeans arrived, during the late eighteenth century, language, culture, and beliefs were similar throughout the area. Tonga is the only country in the Pacific region that has never been colonised, and this is a source of great pride. As a result, there are relatively few immigrants in Tonga, and the claim that Tonga is the “purest” Polynesian society has some justification. Over the last two centuries, Tongans have made a conscious effort to preserve much of the traditional Tongan culture, while adapting some of the western and Christian values introduced by the missionaries. King Tupou I (1797–1893), the founder of modern Tonga, is credited with the resulting synthesis between traditional and western cultural norms, and since his time, it has been referred to as Anga Fakatonga or “The Tongan Way”. Much has been written about the “Tongan Way” (Campbell, 1992; Morton, 1996; Taufe'ulungaki, 2011; and others), and I will suggest that a number of facets of the traditional Tongan culture can help to explain the phenomena described in this article.

As part of the background, it should be noted that over the last 150 years, Tongans have shown an aptitude for mathematics. The early missionaries commented on how successful their students were in algebra and geometry (Moulton, 1921), and today, a considerable number of Tongan students go on to get advanced degrees in Mathematics.

2.2 Language, culture, and mathematical thinking

Rich literature exists on the relationship between thinking and language and also between thinking and culture. The idea that language shapes people’s view of the world has a long history going back at least as far as Roger Bacon in the thirteenth century and Wilhelm von Humboldt in the nineteenth century (Gumperz & Levinson, 1996). The idea was popularised in the mid-twentieth century by Edward Sapir and his student Benjamin Whorf and is commonly known as the Sapir-Whorf hypothesis (Whorf, 1959). Initially there was great interest in the hypothesis, yet, after empirical studies in which researchers tried to test the hypothesis proved inconclusive, its popularity abated, and the original hypothesis has been largely discredited. Whorf’s deterministic approach, that our mother tongue restricts how we think and prevents us from being able to think certain thoughts, has been overwhelmingly rejected. Recently, however, there has been a renaissance of interest in how language and understanding are related (Cole, 1996; Deutscher, 2010), and the dominant more nuanced approach, known as “linguistic relativism,” is “that when we learn our mother tongue, we acquire certain habits of thought that shape our experience in significant and often surprising ways” (Deutscher, 2010). One explanation for the fluctuations in popularity and contradictory findings regarding linguistic relativity is a lack of clarity with regard to the terms in which the hypothesis is stated such as “thinking” or “understanding.” In Section 2.4, I will attempt to give a clear definition of the hypothesis to be tested.

The idea that mathematical thinking is closely related to language and culture is advocated by Saxe (2012) and by Barton and Frank (2001), amongst others. Barton and Frank suggest that “Recent interest in how anthropology and linguistics relates to mathematics has led to recognition that mathematical thinking is a function of language in ways not previously recognised” (ibid., p. 1). They go on to suggest that in many places, words relating to mathematical ideas “do not have equivalents when translated into English. These concepts have referential domains which only partially overlap, or which are non-commensurate, with those of English” (ibid., p. 4). As a result of his research, amongst the Oksampin communities in Papua New Guinea, Geoffrey Saxe developed the idea that not only language but culture and history are related to how mathematical ideas are understood. He proposed a methodological approach “rooted in the idea that both culture and cognition should be understood as processes that are reciprocally related, each participating in the constitution of the other” (Saxe, 2012, p. 16). My research in Tonga originated with the idea that language affects understanding; however, it quickly became clear that the cultural and historical background of the community had to be taken into account.

2.3 The development of probabilistic thinking

In this section, I give a brief summary of the historical development of probabilistic thinking as well as theories about how probability is learnt. Probability, as we understand it today, is a relatively recent development which emerged during a short period in mid-seventeenth-century Europe. The novel idea that uncertainty can be measured on a scale from zero to one answered the needs of a complex society. The historian Ian Hacking suggests that “the emergence of probability was a change more fundamental than any revolution. A new thinking cap” (Hacking, 2006, p.xix). He suggests the following preconditions for the emergence of probability:

• The development of a vocabulary for situations of uncertainty

• The use of probability in assessing evidence for past and future events

• Changing attitudes to religion, fatalism, and causality

• Calculations concerning games of chance

• Developments in the insurance industry and the theory behind annuities

Following the seminal work by Piaget and Inhelder (1975), much has been written about how probabilistic thinking develops. However, almost all of the literature presents this issue in an ethnocentric context concentrating on western culture, and little has been written on these processes in other cultures.

Two surveys of research concerning probability education were written by Shaughnessy (1992) and Jones (2007). Jones raises concern at the lack of cross-cultural studies and “the lack of probability research outside western countries” (ibid., p944). Shaughnessy includes in his “wish list” of future research the need for “cross cultural comparison studies” (Shaughnessy, 1992, p. 489). This can be seen as a motivation for the research described in this paper.

I have only located three articles which do deal with cross-cultural studies on understanding probability. The first, by Amir and Williams (1999), focuses on two cultural groups within the same school. One group is described as being of “English origin” and the other of “non-English origin”. The second study, by Meaney et al. (2012), deals with the difficulties of teaching probability in a Maori speaking school in New Zealand. They report that though “Maori had words to express quantity, space, shape, and measurement, there were very few words to express probability. Traditionally, events were influenced by the mana (power) of certain individuals and were not seen as random events that were more or less likely to occur” (ibid., p. 29). The third, by Chassapis and Chatzivasileiou (2008), focuses on conceptions of chance and probability held by Greek, Jordanian, and Palestinians children. All three articles describe how language, beliefs, and experience influence the children’s “informal knowledge” of probability, which is defined as “the intuitive knowledge they bring to school and use in thinking about probabilistic situations” (Amir & Williams, 1999, p. 85).

The only specific reference which I found about learning probability in Tonga is in Stan Manu’s doctoral thesis about bilingual mathematics education (Manu, 2005). Manu remarks that probability is one of the subjects with which teachers have difficulties:

With regard to the concept of probability, for example, one teacher in this study recounted how difficult it is for him to teach and distinguish between the meanings of non-equivalent terms, such as ‘very likely’, ‘probable’, and ‘almost certain’, when these concepts have to be expressed and explained in Tongan. (ibid., p. 64)

Mention should also be made of a fascinating book by David Lancy (1983), titled Cross-Cultural Studies in Cognition and Mathematics. The author does not deal specifically with probability but describes in detail a multi-cultural study of how mathematics is learned in different communities in Papua New Guinea. The aim of the project was to “examine systematically the relationship between the cultural background of the child and his or her pattern of cognitive development and acquisition of school arithmetic” (Lancy, 1983, P. xv).

2.4 Conceptual framework for this study: commognitive approach

In this study, I adopt the discursive approach, in which mathematics is defined as a form of communication or discourse (Sfard, 2008). This approach is referred to as commognitive, and as the name implies, cognitive processes and communicating are understood as different instances of the same phenomenon. A community of discourse is defined as those individuals who are capable of participating in any given discourse.

The commognitive approach to mathematical discourse includes a number of definitions and concepts, and the most relevant for this paper include:

• Endorsed narratives: A “narrative that is regarded as reflecting the state of affairs in the world … the narrative is consensually endorsed by the community of the relevant discourse” (Sfard, 2008, p. 298).

• Discursive routines: Routines which are patterns that are defined by sets of rules for how to perform a given task and when to perform it.

• Community of discourse: Those individuals capable of participating in any given discourse and by the endorsed narratives which they use.

• Sfard differentiates between two kinds of learning:

  • “Object-level learning”—the expansion of the existing discourse, extending vocabulary, new routines. and a change in the set of narratives already endorsed.

  • “Meta-level learning”—changes in the meta-rules. The change here is from one discourse (set of words, mediators, and routines) to another. In most cases, meta-level learning happens when someone encounters a new discourse governed by a different set of meta-rules.

• “Commognitive conflict” is associated with meta-level learning and is caused by conflicting sets of meta-rules which might use the same words in different ways. This leads to different participants endorsing incommensurable discourses.

The commognitive approach provides a fully developed theory of “discursive” learning based on the underlying assumption of the unity of thinking and communicating. This provides the operational definitions necessary to make the idea of linguistic relativity empirically testable. Communication does not have to be in words and may be with others or with oneself. In this latter case, it is called thinking. The idea behind linguistic relativity can thus be restated as “Language moulds thinking (discourse)” (Sfard, 2008, p. 100).

3 Research questions and methods

3.1 Methods for comparing Tongan students’ access to probability in comparison with other mathematical subjects

Before attempting to explain the puzzling phenomenon that inspired this study, it was necessary to provide evidence that my students’ special struggle with probability was not an isolated incident but part of a wider picture (RQ1). In other words, I needed evidence that Tongans have greater problems understanding probability than they do when learning other subjects in the mathematics curriculum. The evidence is presented in three parts:

• A simple written test was designed to compare students’ ability in four fields of the mathematics curriculum: algebra, geometry, probability, and fractions (shortened forms of the questions can be found in Table 2 in Section 4.1.1). The test was given to 56 students in Tonga, and the same test was given to a comparison group of 28 Israelis of the same age. In both cases, the students were in their last 2 years of compulsory mathematics education. In Tonga, all the questions were written in both English (the official language for teaching at the secondary level) and in Tongan. In Israel the same questions were translated into Hebrew. A serious attempt was made to insure the equivalence of the questions in the three languages used (Tongan, English, and Hebrew) and that translation would not justify different responses. The questions were designed to test if the Tongan students show a relative strength or weakness in any of the subjects tested. The questions were relatively straightforward for a western educated student in the last years of high school. The test consisted of three questions in each of the four categories (Algebra, Geometry, Probability, and Fractions), giving a total of twelve questions. Each of the answers was graded as correct or incorrect, so every student could score between zero and three for each of the given subjects.

• An analysis of exam reports: Tongan students take two sets of public exams. The Tongan School Certificate (TSC) is taken by 16-year-olds and is set by the Tongan Ministry of Education, while the form seven exam (taken by 18-year-olds) is administered from Fiji and is set by the South Pacific Board of Educational Assessment (SPBEA). In both cases, examples of the examiners’ assessments regarding mathematics exams were obtained. These were used to identify specific subjects in which Tongan students were reported to have difficulties.

• Interviews with teachers: I interviewed a number of mathematics teachers from various primary and high schools to learn from them which subjects on the mathematics curriculum were particularly difficult for their students.

3.2 Methods used to identify the Tongan discourse about uncertainty and the probability of future events

A central part of the research was a series of sixty semi-structured interviews which were carried out with the aim of providing empirical evidence concerning the discourse on uncertainty in Tonga (RQ2). All the interviews were audio-recorded, and some were video-recorded. Where necessary a translation was provided by a local research assistant. The interviews were carried out in public places (markets and shopping centres), and we aimed to recruit participants representing a cross section of Tongan society, in terms of age, sex, religious affiliation, and employment. The participants were not selected in a randomised way, but some of the questions were designed to confirm that the respondents were broadly representative of the whole Tongan population, as shown in Table 1. The respondents were all asked if they were willing to participate, and there was a very low refusal rate, which reduced the chances of a skewed sample.

Table 1 Cross section of the sixty respondents included in the semi-structured interviews

The interviews helped to focus on the endorsed narratives concerning uncertainty and how those narratives observed in Tonga differ from those in the West, what the differences are, and what cultural and linguistic phenomena underpin those differences.

The results of the semi-structured interviews were used to identify which subjects were to be pursued in the following separate sub-studies:

• An analysis of the vocabulary available for situations of uncertainty in Tongan. This was achieved by interviewing experts on the Tongan language as well as analysing the vocabulary in various dictionaries.

• Classroom observation including a series of lessons on probability at a high school.

• A series of in-depth conversations with children and adults. Most of the conversations were with groups of two or three. The children were recruited from local schools and the adults from women’s groups and other people to whom I was introduced.

• In-depth interviews with various professionals including bankers, insurance agents, lawyers, the minister of education, language experts, bingo players, and church ministers. These were in addition to the sixty semi-structured interviews.

3.3 Methods of data analysis

The research was partly quantitative and partly qualitative, and this was reflected in the data analysis.

The classroom tests and the semi-structured interviews provided quantitative evidence for differences between different groups and populations. The only statistical tools used were Student’s t-test, the results of which were supported using the Wilcoxon non-parametric test. The tests were carried out using the statistical platform “R”.

All the interviews and classroom observations were audio-recorded, and some were video-recorded. Some, but not all, of the interviews were transcribed.

4 Findings

4.1 Evidence that the discourse on probability is less accessible to Tongan students than other mathematical discourses

4.1.1 A simple written test

The results of the simple written test which was designed to compare the ability of Tongan and Israeli students, of a similar age, in algebra, geometry, probability, and fractions are presented in Table 2.

Table 2 A comparison between the grades of Tongan and Israeli students in four fields of the mathematics curriculum

These results provide strong evidence that while Tongan students are proficient at algebra and geometry (relative to similar students in a western country), they have very significant difficulties answering what are seen in the west as simple problems concerning probability and fractions. It should be noted that a separate, parallel part of the research concerned the discourse on fractions in Tonga, but there is no room to describe the results in this paper (Morris, 2017). The two subjects are, of course, related since the conventional way of measuring probability is to use fractions between zero and one.

4.1.2 An analysis of exam results

The examiners’ assessments of the public exams provided further evidence that Tongan students have specific problems when answering questions concerning probability. Typical examples were as follows:

Question ten [the only question on probability]: Poorly done. Basic skills of drawing probability tree and assigning probabilities is a major problem. (TSC, 2009)

Probability: This topic has been poorly answered since the beginning of the SPFSC examination. (SPBEA, 2007)

4.1.3 Interviews with teachers

The research included a number of interviews with mathematics teachers at primary and secondary level. All three interviewed secondary teachers stated that probability and statistics are particularly difficult subjects to teach. A typical comment was made during an interview (in English) with a teacher at one of the most prestigious high schools. When asked which subjects the students find particularly difficult, he responded that “It is very hard to teach the statistics and probability because there are so many things to explain and read, it is different from calculus, where you just have to write things.” He also suggested that the lack of vocabulary in Tongan makes teaching statistics hard: “I know there are a lot of readings in statistics like the word problems … only those who are really good in English, they like the statistics.”

4.2 Tongan discourse about uncertainty and the probability of future events (RQ2)

The findings described in this section deal with a collection of different aspects of uncertainty and probability. In order to give some structure to the findings, they are organised so as to reflect some of the necessary conditions for the emergence of probability in seventeenth-century Europe (outlined in Section 2.3 above). These include relevant vocabulary, estimating the probability of future events, estimating levels of uncertainty, attitudes to religion and fatalism, games of chance, and developments in insurance.

4.2.1 The vocabulary for uncertainty

Having defined the hypothesis of linguistic relativism as “Language Moulds Thinking,” it was important to look at the language available. An important starting point in identifying the endorsed narratives concerning uncertainty is an analysis of the commonly used vocabulary relating to the subject. In English there is a large selection of words which express various levels of uncertainty as shown in Table 3. One of the observations that motivated the research was the difficulty that I had when trying to translate words for levels of uncertainty from English into Tongan. I encouraged my students to translate ideas and words, but when it came to terms such as “very likely,” “unlikely,” or “almost certain,” my students could not find suitable vocabulary. They kept returning to the one word “Mahalo.”

Table 3 A sample of English vocabulary for different levels of uncertainty (the percentages are included to give a rough idea of the level of uncertainty)

In order to try to understand how levels of uncertainty can be and are expressed in Tongan, I looked at formal definitions using three commonly used dictionaries (Table 4). I also asked Tongan language experts to provide me with accurate translations. In addition, I attempted to find out how people use and understand the available vocabulary.

Table 4 Translations of vocabulary concerning levels of uncertainty in three commonly used dictionaries

The two words which appear most are mahalo (also mahalo pe) and ngalingali. No translation is given for the English word “maybe”, but one dictionary refers the user to “perhaps.” Ngalingali can best be translated as “likely,” whereas mahalo is best translated as “perhaps.” I asked various Tongan language experts to give the correct translations for various English words, and they confirmed the dictionary definitions. When asked for translations of other English phrases or words relating to levels of uncertainty, the same words listed above were repeated.

During our interviews, we tried to find out what vocabulary is used in practice, and it became clear that mahalo and mahalo pe are the most commonly used and best understood words for expressing uncertainty. Ngalingali is generally understood but is not used by most people. For example when asked “what is the chance that it will rain during the next month?” (something which is likely but not certain), only 10 out of 59 respondents used ngalingali [likely], while 21 answered mahalo [maybe/perhaps] and 17 pau 'aupito [certain].

Further evidence for the paucity of vocabulary describing situations of uncertainty was found in the official mathematics curriculum. The curriculum for middle schools¹ is written in both Tongan and in English with parallel texts on opposite pages. This is true of the whole mathematics curriculum except for the section on statistics and probability, which is written in English without a translation into Tongan. In an interview at the Curriculum Development Unit, I asked why the curriculum for statistics and probability did not include a Tongan translation. It was explained that “the Tongan was not included because suitable vocabulary does not exist in Tongan to translate the English syllabus.”

The mathematics curriculum for primary schools (TCDU, 2009) is written entirely in Tongan except for the section on statistics and probability, which is written in Tongan and in English. The curriculum includes a brief dictionary of English words concerning probability and uncertainty together with their Tongan equivalent. When I showed this dictionary to Tongans, they informed me that they would never use it and would not understand many of the Tongan words. Some examples are shown in Table 5.

Table 5 Examples of translations of words concerning probability from the primary school curriculum

The above findings are reinforced by an observation in Stan Manu’s doctoral thesis about bilingual mathematics education in Tonga:

With regard to the concept of probability, for example, one teacher in this study recounted how difficult it is for him to teach and distinguish between the meanings of non-equivalent terms, such as ‘very likely’, ‘probable’, and ‘almost certain’, when these concepts have to be expressed and explained in Tongan. (Manu, 2005, p64)

In summary, it is clear that there is a very limited Tongan vocabulary for levels of uncertainty, and it follows that the discourse on uncertainty and probability in Tonga will not be commensurable with the canonical western discourse.

4.2.2 Estimating the chances of future events

The respondents were asked to estimate the chances of various future events happening. The questions included:

• What is the chance that it will rain tomorrow? [Ko e ha e faingamalie 'e 'uha 'apongipongi?]

• When tossing a coin, what is the chance that it comes up heads? [Oku ou vilohi seniti 20. Koe ha e faingamalie e 'asi hake koe 'ulu?]

• A couple is expecting their first child. What are the chances (in percentage) that it will be a boy and not a girl? [Okapau 'oku teu fanau ha ongomatua ko ha e peseti 'oku ngalingali ko e tamasi'i pe ta'ahine?].

There is not enough room here to list all the varied responses to all the questions, so I include a detailed description of the answers to the question concerning rain and a summary of the responses to the other questions.

Before answering the question concerning the chances of rain, almost all the respondents looked up at the sky, presumably to look at the clouds. All sixty responses are summarised in Table 6 in which I have attempted to classify the answers as either “certain” or “uncertain.” Thirty-five out of sixty respondents answered with what I classified as a “certain” answer, fifteen avoided giving a straightforward answer, and only ten out of sixty gave answers in terms of “uncertainty,”

Table 6 Responses to the question “What are the chances that it will rain tomorrow”?

The answers to the question concerning coin tossing and the chances of giving birth to a boy showed a similar pattern. In each case, less than twenty-five percent of the answers were in terms of uncertainty. All the other answers were in terms which to my Eurocentric way of thinking may appear dogmatic. The responses included:

The chances of a head:“You will know when you turn it over”

“Depends who tossed the coin”

“Toss it four times to get a head”

The chances of a boy:“I believe they will get what they wish for”

“Depends on the scan”

“Sleep on time, wake up on time [then it will be a boy]”

In order to understand the discourse underlying the responses, a number of respondents were asked how they would feel if the outcome was not what they had predicted (e.g., if it will not rain tomorrow). A typical response was:

A ‘maybe’ answer would be dishonest. A definite answer is more honest. Not good to have doubts when you answer question. If you have doubts that is bad. If you ask the child and he answers ‘maybe’ the child will get a slap (you are being cheeky) [...] I feel good that I am positive about something that I believe in, I didn’t have doubts.

The answers to all the above questions indicate that the discourse in Tonga stresses predicting the future in terms of certainty and that uncertainty about the future is not well respected. What clues do the answers to these questions give us when trying to understand what underlies the way in which uncertainty is understood?

Though the questions asked were about the chances of a future event, most of the respondents seem to have understood that they were expected to give a prediction. The question “what is the chance that ...? (Koe ha e faingamalie e ... ? )” does not seem to evoke any association except that one needs to say something about the result of a future event. It would seem that there is no routine for speaking about chances and for completing a sentence such as “the chance of such and such an event is... .”

The responses provide evidence that future events are understood to be dependent on previous actions or information and are in some sense predetermined. For example, the sex of the baby depends on the scan, the outcome when tossing a coin depends on who tossed the coin, and the weather tomorrow depends on the clouds today. This observation led to further questions regarding the certainty of future events and predetermination, which will be examined below.

4.2.3 Estimating levels of uncertainty

The participants were given a chart showing the following spectrum of words (in Tongan and English), defining increasing levels of certainty, and were asked to place various events on the spectrum:

• ImpossibleTa'e pau

• ImprobableNgali ta'e pau

• MaybeMahalo

• LikelyNgalingali, Hange

• Almost certainMeimei pau

• CertainPau'aupito

Considerable care was taken to choose the correct translation of the English words, but with hindsight, some of the words are not often used or well understood (as far as I could ascertain, no other commonly used words are available in the Tongan vocabulary). This is significant in itself, and, as well as explaining the responses that were recorded, it can help to explain the difficulties that Tongans have in expressing levels of uncertainty.

The events were chosen with the aim of ascertaining what spectrum of answers the respondents would give. They were asked to place the following list of events on the scale:

• a) Li e taisi pea 'asi hake 6 [Throwing six with a dice].

• b) Ihe ta'u 10 ka hokomai 'e palemia 'a Akilisi Pohiva [Akilisi Pohiva (the leader of the opposition) will be prime minister within the next 10 years].

• c) E lava 'o puna e puaka [Pigs can fly].

• d) Te tau ongo'i ha mofuike 'i Nuku'alofa 'apongipongi [We will feel an earthquake in Nuku'alofa during the next 24 h].

• e) 'E 'uha tu'o taha pe lahi hake 'i Nuku'alofa he mahina kaha'u [It will rain at least once in Nuku'alofa during the next month].

• f) 'E ha'ele mai a kuini 'Elisapeti ki Tonga he ta'u kaha'u [Queen Elizabeth will visit Tonga next year].

The responses are summarised in the following graph, and as can be seen, very few respondents chose “improbable,” “almost certain,” or “likely,” which supports the claim that the words which were suggested (ngali ‘ikai, meimai pau, and ngalingali, respectively) are not in common use and are not well understood. The answers were concentrated in the extremes (“certain” and “impossible”) and in the middle (“maybe”).

There was a tendency to give what might be seen as dogmatic answers (certain or impossible). For example, when asked the chances of throwing a six with a die, 19 out of 59 answered “Ikai Malava” [impossible]. The participants were given a die, and I believe that they understood the question (Fig. 1).

Fig. 1

Graph showing spread of answers to questions about uncertainty. (6 questions were answered by 59 Tongan respondents giving a total of 354 responses)

4.2.4 Attitudes to religion, fatalism, and causality (responses concerning probability according to church affiliation)

The structured interviews included background questions such as sex, age, years studied abroad, and church affiliation. This information was used in part to check that we had interviewed a reasonable cross section of Tongan society, but it also gave an opportunity to test if there are significant differences between the various groups. This was tested by giving each of the participants a score reflecting their answers to questions about probability. The scores were designed to highlight answers which were different from the expected western responses.

No significant differences were found between respondents according to age or sex. I had expected a significant difference between those who had spent years educated abroad and those who had not, but no such difference emerged. However, what came as a great surprise was the difference between the scores of the members of different Christian denominations.

Tonga is a deeply Christian country, and church affiliation is part of everyone’s identity. The largest denomination is the Wesleyan Church followed by the Mormon and Catholic churches. As can be seen from Table 7, the scores of the Wesleyan respondents were significantly higher than those scored by the Catholics and the Mormons. A p-value of 0.0074 provides strong evidence that the answers given by Wesleyans were significantly closer to the canonical western responses.

Table 7 The results of a t-test to compare probability scores between different denominations

These results came as a surprise and motivated an investigation regarding attitudes to religion, free will, and predetermination and their effect on how uncertainty is understood. It emerged that Wesleyan theology stresses the existence of free will. As Wesley stated in one of his sermons:

If man were not free, he could not be accountable either for his thoughts, word, or actions. If he were not free, he would not be capable either of reward or punishment; he would be incapable either of virtue or vice, of being either morally good or bad. If he had no more freedom than the sun, the moon, or the stars, he would be no more accountable than them. (Wesley, 1835, p39)

In contrast the theologies of both the Mormon and the Catholic Churches are more equivocal and inconsistent regarding free will. Both churches are based on a strictly disciplined, hierarchical structure. The Catholic Church is headed by the Pope and includes the concept of “Papal Infallibility”. The President of the Mormon Church is regarded as a “Living Prophet” and is the spokesman of God on Earth.

How does all this affect how Tongans understand free will and predestination? I interviewed a number of religious leaders to ascertain how they saw this question and though critical of the prevailing attitude, they confirmed that there is a dominant fatalistic attitude in the Christian community in Tonga. A typical explanation of a Wesleyan minister was:

There is a fatalism, there is a simple faith ... I think it is just embedded in their particular Christian faith and when they pray they almost invariably start off with Otua Mafi Mafi, Almighty God. I never pray that way. I can’t use that language it is just anathema to me.

He went on to suggest that this fatalistic attitude was linked to the traditional pre-Christian attitude of obedience to the chief and the importance of unquestioning respect.

We asked a number of people about the role of God in deciding future events. Most people gave answers that might be seen, from a western perspective, to be confused answers. For example:

Interviewer: “Does God decide when our time will be or do we control our future?”

Respondent: “God decides our time, why, because God is the ruler of our lives”

Interviewer: “Are we responsible for a decision to smoke cigarettes?”

Respondent: “Yes it is our responsibility to choose”

Interviewer: “If you die of lung cancer then who is responsible for dying young”

Respondent: “It is God’s decision but God gave a freedom to decide, like I said before God is the ruler of our lives, but it is our choice and decision but God is the ruler”

In summary I found strong evidence that religious belief as well as traditional attitudes to obedience are associated with an equivocal attitude to free will and a certainty about the future—this can have implications about how uncertainty and probability are understood.

4.2.5 Betting and games of chance (bingo)

Where they exist, betting and games of chance provide an important introduction to the ideas of probability, but we found no evidence of betting or gambling in Tonga, with the one exception of Bingo.

As well as participating in a few games, I interviewed some of the participants in order to ascertain how they understood the uncertainty involved in Bingo. The participants gave seemingly contradictory answers regarding the randomness of the numbers called. At the same time as recognising that the numbers were called at random, they had various strategies based on lucky cards or lucky days. These observations were not substantially different from those made in Bingo halls in the west (see, e.g,. Griffiths & Bingham, 2005).

I found no evidence of the type of gambling (such as Poker or horse racing), which might lead to a demand for a probabilistic assessment of various possible strategies, as happened in seventeenth-century Europe². In contrast Bingo is a game based entirely on chance, and there is no practical use of calculating probabilities of various strategies. Thus it is not surprising that Bingo players did not appear to be interested in making calculations regarding the probability of winning.

Gambling remains illegal in Tonga, but in recent years, there have been suggestions of building a casino to attract foreign tourists, and there is evidence that online gambling is becoming more common. It can be expected that these developments will affect the use and understanding of probability and encourage the teaching of the subject.

4.2.6 Attitudes to insurance and planning for the future

As part of my field work, I interviewed a number of professionals including bank managers, the director of microfinance in Tonga, the minister of education, and a school principal. A consistent picture emerged of a society in which planning for the future is not understood in the way that it is in the west.

In an interview with the minister of education, I asked how Tongans think about the future. She responded:

I think they are quite certain that the processes and the checks and balances we have in the society insure that the future will be taken care of. Of course there are so many things happening these days, the future is pretty uncertain, but the Tongans don’t think of it that way. They think that they have their insurance [which] is their children [and] their extended family. Of course, these have a tendency to break down these days, children migrating, …. so they will have to start thinking about the future seriously. If you ask a Tongan to pay out for life insurance, they will think it is a total waste of money – I agree.

I asked the principal of a school what the attitude of Tongans is towards planning for the future and pensions. His response was:

Tongans don’t actually live for tomorrow, they live for today. Look out for today don’t think about consequences, eat all you have and give it away, don’t save for tomorrow. […] No [there is no need for insurance] the family will take care of them [....]. We will just trust in God. Even with health, God will call you when it is time to go.

The above examples, together with many other conversations and interviews, provide evidence for the claim that attitudes about the future and planning, if they exist at all, are very different from those in a typical western country. At the same time, they gave some evidence (in particular the interview with the minister of education) that a process is underway, whereby the discourse on probability and uncertainty is changing or will be forced to change.

5 Discussion

The findings described in Section 4.1 provide a clear positive answer to the first research question. At least for the small sample studied, there is overwhelming evidence that the discourse on probability is less accessible to Tongan students than other mathematical discourses (RQ1). It should be stressed that this is a specific problem concerning probability and that Tongan students have a good reputation regarding general mathematical ability.

The findings presented in Section 4.2 provide a number of clues concerning the second research question: “What is the Tongan discourse about uncertainty and the probability of future events?” (RQ2). In the remainder of this section, I use these clues to attempt to solve the puzzle which motivated the research.

5.1 An attempt to understand the observed discourse on uncertainty and probability

The picture which emerges is of a society which, historically, has had little need for assessing degrees of uncertainty. Definite answers are valued above uncertain answers, and this could be linked with fatalism about the future. The picture is in many ways similar to that described by Hacking in seventeenth-century Europe (Hacking, 2006). As a result of contact with western countries, the ideas of probability are beginning to emerge together with the need for insurance, changing attitudes to religion and predestination, and the need to plan for possible future events (Morris, 2017).

Since the reign of Tupou I, in the mid-nineteenth century, the Tongan way of life and value system have been dominated by what is known as “The Tongan Way”, which aims to combine traditional Tongan values of respect and obedience with a deeply held Christian belief. In my research, I have explored how this belief system affects and is affected by attitudes to free will, predetermination, and the inevitability of future events. Most of the people whom I interviewed expressed a strong belief that future events depend on the will of God and that we do not have free will to control them. I found evidence that the importance of obedience to God and to those of higher status goes hand in hand with a negative attitude to expressions of uncertainty.

How does all this help us to understand the Tongan discourse on uncertainty? I attempt to answer this question by referring back to one of the anecdotes which motivated the research (described above), the incident in which one of my students was certain that it would be sunny at the end of the lesson tomorrow because “the angels told me”. What was the way of thinking behind my student’s response? Let me indulge in speculation and propose the following sequence of endorsed narratives:

• God is all powerful [Otua Mafi Mafi].

• The future is predetermined because God is all powerful.

• It follows that there is no such thing as uncertainty regarding whether it will be sunny at this time tomorrow. It is no more uncertain than whether it was sunny yesterday.

• Thus, any discourse about future events is a discourse about definite events, which are known to God.

• When asked the chance that it will be sunny tomorrow, there is no need for vocabulary describing uncertainty. Any question about what will happen in the future is a question about what God has predetermined.

• Given all the above, what is the better way to answer the question about what God has decided than to go through the intermediary of “the angels”?³

Unlike students in a typical western classroom, my students were faced with ideas which had no basis in their day-to-day discourse. The endorsed narratives with which they had grown up were not relevant to the concepts of probability as it is understood in the west and as it appears in the mathematics syllabus. Using the terminology of commognition, my students faced a “commognitive conflict,” a conflict between the traditional Tongan attitudes to uncertainty and the demands of the wider world (and of their teachers) to join the dominant community of discourse.

5.2 An example of linguistic relativism

The study was based on the expectation that discourses on uncertainty would prove to be particularly promising settings in which to test the ideas behind linguistic relativism. My original hypothesis was that the everyday, informal vocabulary (or lack of it) for levels of uncertainty affects people’s ability to grasp the relevant formal mathematical ideas. However, as the research progressed, I came to the realisation that the question is not of cause and effect (does language affect thinking?) but of the mutual relationships between cultural norms, language, and thinking. Therefore, I adopted the specific version of the hypothesis that “language moulds thinking (discourse)” (Sfard, 2008, p100). Using this approach, I have brought evidence that cultural norms, together with language, define the endorsed narrative which I observed in Tonga. Attitudes to uncertainty, fatalism, free will, obedience, and predetermination combine with a dearth of vocabulary to provide a lens through which probability is understood.

5.3 A more modest mathematician returns from the South Pacific

The study presented here provides a strong evidence that different everyday discourses lead to a different understanding of basic ideas concerning uncertainty. The research described was carried out in one small island country, but I suggest that there is a need to look at the discourse on uncertainty and how it should affect the teaching of probability in the mainstream western system as well as in developing countries other than Tonga and in particular in countries where the medium of education is not the students’ mother tongue. In the Second Handbook on Mathematics Teaching and Learning, Jones states that “We were not able to locate cognitive research on the subjective approach to probability measurement” (Jones, 2007, p. 925). I hope that my research might be a first step of a new way towards filling that gap.

My conclusions lead to a dilemma concerning contradictory aims of education in a developing country:

• On the one hand, I have come to realise that there is not one correct discourse concerning uncertainty and probability. The everyday discourses typical of various communities, such as the one I found in Tonga, deserve to be understood and respected. What right do western or Tongan educators have to replace the given endorsed narrative with the dominant western one?

• On the other hand, the dominant western discourse of uncertainty, chance, and probability has proved to be overwhelmingly successful. Probabilistic thinking (in the western sense) has become an essential skill in the modern world. In order to supply people (in our case from Tonga) with these desirable tools, it is necessary to transform their discourse. Using the vocabulary of the commognitive approach, this will involve a “commognitive conflict” between different “endorsed narratives.”

It is not easy to find a synthesis between these two opposing approaches. I would suggest that a serious attempt be made to understand and respect the local discourse, which is to be transformed, and at the same time to respect the wishes of the people involved who, as far as I could ascertain, do want to learn how to use the tools offered by the western discourse. In the Tongan context, this can be seen as part of an ongoing attempt to reinforce and reinterpret the Tongan Way, which aims to find a synthesis between traditional Tongan values and useful western ideas. This reflects the course set by King George Tupou I, the founder of modern Tonga. In a speech made to the legislative assembly in 1882, he stressed the importance of education and added:

If there be anything in foreign lands which will be useful to us, it is right for us to desire to get it; but it is also right if there is any Tongan custom which is useful, for us to preserve it. (Campbell, 1957, p69)

It seems appropriate to give the last word on this subject to 'Ana Maui Taufe'ulungaki, who has been the dominant figure in the move to incorporate more Tongan values into the education system over the last 30 years and was, until recently, able to implement her ideas as the Minister of education. Referring to the Tongan values and the Tongan Way, she stated:

The key values of western societies are often said to relate to individual rights and freedoms; justice in terms of equity and access; protection of privacy; promotion of competition and consumerism; and, scientific-rational thinking. Tongan values, on the other hand, which are similar to the values of other Pacific communities, emphasise the holistic nature of life and the centrality of good relationships; the connectivity of the past, present and future; of people, land, sea, and sky, and the spirituality that bind them together. (Taufe'ulungaki, 2011)

A new kind of education was evidently called for: it would be an education that would truly reflect the ideals and values of Tongan society; an education that would offer a meaningful preparation of young people for the life they would be expected to lead in the community; and an education designed to develop in young people realistic attitudes to life, commensurate with the Tongan socio-economic context. (Taufe'ulungaki, 1979, p26)

I arrived in Tonga with a clear idea of how statistics and probability should be taught and understood. My observations in Tonga led to a slow transformation of my ideas. I learnt from my Tongan students, colleagues, and friends that there is not one all-embracing approach to uncertainty and probability but that they can be understood in radically different ways by different people and in different cultural contexts.