Worldviews, religions, and beliefs about teaching and learning: perception of mathematics teachers with different religious backgrounds

Worldviews, religions, and beliefs about teachingand learning: perception of mathematics teacherswith different religious backgrounds

Yip-Cheung Chan & Ngai-Ying Wong

# Springer Science+Business Media Dordrecht 2014

Abstract Beliefs about mathematics education and their influences on teaching practices havebeen widely investigated in recent decades. There have been numerous empirical studies on theinfluences of religions on teachers’ and students’ beliefs about subjects such as sciences andlanguage. However, the influences of worldviews in general and religions in particular, as oneof the major sources of beliefs in relation to mathematics education, are under-researched. Thecurrent study is a first step to unpacking the relationship between teachers’ religions and theirbeliefs about mathematics teaching and learning. By means of semi-structured interviews withmathematics teachers of different religious backgrounds, teachers’ perceptions on the connec-tion between their personal religious beliefs and their beliefs about teaching and learning areinvestigated. In-depth analyses of the perceptions of three mathematics teachers reveal thecomplex relationship between teachers’ religious beliefs and their teaching beliefs. First, thereare some common values shared by different religions, which influence the beliefs aboutmathematics teaching and learning as well as education in general. Second, religion is a richbelief system, and the teachers appear to apply only a portion of their religious beliefs to guidetheir teaching. It is also possible that a teacher is influenced bymore than one religion or culturaltradition. Despite its subtleties, our study provides evidence to support the alignment betweenteachers’ personal religious beliefs and their beliefs about mathematics teaching and learning.

Keywords Beliefs aboutmathematics teachingand learning .Values .Personal religiousbeliefs .

Worldviews . Case study

1 Teachers’ worldviews and beliefs about mathematics and mathematics education

How teachers’ worldviews, beliefs1 and values might influence their students has been aneducational research focus in recent decades (Zhang & Wong, 2014). In this paper, we use the

Educ Stud MathDOI 10.1007/s10649-014-9555-1

1Different terminologies such as “conception”, “belief ”, “view”, and “image” have been used in the literature.Although it has been argued that there may be subtle differences (Philipp, 2007), we use these terms inter-changeably in our study.

Y.<C. Chan (*) :N.<Y. WongDepartment of Curriculum and Instruction, The Chinese University of Hong Kong, Hong Kong, Chinae-mail: [email protected]

definitions of the terms “beliefs” and “value” of Philipp (2007). Accordingly, beliefsare “psychologically held understandings, premises, or propositions about the worldthat are thought to be true.… Beliefs might be thought of as lens that affect one’sview of some aspect of the world or as dispositions toward action” (Philipp, 2007, p.259). A person’s beliefs are rather complex and may consist of different clusters.Some belief clusters are more deeply held than others, which forms one’s set ofvalues. Value can be defined as “the worth of something [and] a belief one holdsdeeply, even to the point of cherishing, and acts upon” (Philipp, 2007, p. 259).

It is well established that teachers’ beliefs about mathematics and mathematics teaching andlearning play a significant role in their instructional practices (Philipp, 2007; Thompson, 1984,1992; Zhang & Wong, 2014). Yet, such a belief-practice relationship is rather complex. Inparticular, the relationships should be interpreted in terms of contextual factors such as schoolenvironments and cultures (see for instance, Skott, 2009; Speer, 2005). Teachers’ practices inmathematics teaching shape students’ learning experiences which consequently results instudents’ cognitive and affective learning outcomes. The learning outcomes include theirmathematics achievement, problem-solving strategies as well as their beliefs about mathemat-ics and mathematics learning. The whole process is depicted in Fig. 1. In brief, teachers’beliefs about mathematics education are a factor of how students’ learning experiences isformed which cannot be ignored (Wong, Marton, Wong, & Lam, 2002), although we are fullyaware that there are other factors (for instance, teachers’ professional knowledge and schoolcontexts) at play.

Based on the above conceptualisation, a number of studies on beliefs about mathematicsand mathematics teaching and learning have been conducted among both students and teachersin the Chinese regions of the Chinese mainland, Taiwan, and Hong Kong. It was found thatstudents’ and teachers’ beliefs mirror each other: they believed that mathematics involvesthinking and is useful, and mathematics is perceived as more or less a subject of “calculables”.Furthermore, operational procedures, practices, and memorization are found to be central inteaching and learning mathematics. The results of these studies painted a general picture ofeffective mathematics classroom learning and teaching in the eyes of the Chinese, both forstudents and teachers (Zhang & Wong, 2014).

When further investigating teachers’ beliefs about mathematics and mathematics teachingand learning, we realized that there are numerous factors influencing their formation. Teachers’worldviews are one such important factor. For instance, the aim to help students obtain highscores in public examinations may conflict with students’ genuine understanding of mathe-matical concepts, and teachers’ worldviews might influence how they approach this dilemma.Worldview refers to “one’s comprehensive set of beliefs about the nature of reality and how

Teachers’ beliefs

about mathematics

and mathematics

teaching and learning

Teachers’ practices

in mathematics

teaching

Students’ mathematics

achievement and

problem solving

strategies

Students’ beliefs about

mathematics and

mathematics learning

Fig. 1 Teachers’ beliefs and practices and students’ learning

Y.-C. Chan, N.-Y. Wong

one should live in the light of those beliefs” (Heie, 2002, p. 99). Worldview is an integrated“product” of multiple sources such as one’s religious beliefs, academic and general knowledge,personal experience at schools and workplaces, and political ideology. As a non-negligiblefactor of one’s worldview, teachers’ religious beliefs, no matter how strong or subtle they maybe, will have an impact. This paper focuses on the religious origin of worldview and itspossible connection with teachers’ beliefs about mathematics and mathematics teaching andlearning (Fig. 2). Teachers’ practices in mathematics teaching are beyond the scope of thisexploratory paper.

Readers should be reminded that it is oversimplistic to assume that when people subscribeto a certain religion, they possess a unified set of worldviews and act accordingly. Differentbranches of ideology within a single religion evolve over time. (For instance, see Kanitz(2005) for the multiple worldviews of the Christian faith and Wong, Wong, and Wong (2012)for the case of traditional Chinese religions.) At the individual level, one’s interpretation ofreligious beliefs is a “social construct based broadly on the various experiences (and moreparticularly on the religious experiences) that a person lives through” (Mansour, 2008, p.1,608). Religious beliefs are so rich in content that teachers may apply only a portion of themto their (mathematics) teaching. We agree with Mansour (2008) in calling an individualreligious follower’s interpretation of religious beliefs personal religious beliefs (PRB). Theterm is formally defined as “the views, opinions, attitudes, and knowledge constructed by aperson through interaction with his/her sociocultural context through his/her life history andinterpreted as having their origins in religion” (p. 1,608).

The relationship between teachers’ PRB and their beliefs about the nature of mathematicsand mathematics teaching and learning is precisely the theme of the study reported in thepresent paper, which is based on Mansour’s PRB model. While detailed description ofMansour’s PRB model will be given in Section 4, one of its features is highlighting the subtleimpact of teachers’ worldview in general and religious beliefs in particular on their pedagog-ical beliefs (and consequently their teaching behaviors). This study aims precisely to addressthe complexity underlying the interplay.

2 Religions and education

Literature as early as the 1990s has raised the possibility that religious beliefs influenceteaching and learning. The Stevenson team attributed the east-west difference to the effort-ability dichotomy, in which Confucianism was explicitly mentioned (Chen, Stevenson,Hayward, & Burgess, 1995; Stevenson & Stigler, 1992). Later on, Wong et al. (2012)discussed extensively how the traditional Chinese religious beliefs of Confucianism,

Focus of this research study

Teachers’

worldviews and

personal religious

beliefs

Teachers’ beliefs

about mathematics

and mathematics

teaching and

learning

Teachers’

practices in

mathematics

teaching

Students’ mathematics

achievement and

problem solving

strategies

Students’ beliefs about

mathematics and

mathematics learning

?

Fig. 2 Teachers’ worldview, belief and practices, and students’ learning

Worldviews, religions, and beliefs about teaching and learning

Daoism, and Buddhism influence education in general and mathematics education in partic-ular. There have also been voluminous discussions on how ideologies in Western religionsaffect education (see for instance, Heie, 2002; Howell & Bradley, 2001, 2011; Sutcliffe, 2009).In particular, a Christian perspective may suggest that God has given some people the capacityand interest to study mathematics which “is intended to enable us to serve God and otherhuman beings” (Howell & Bradley, 2001, p. 372).

There are also a number of empirical studies about the influences of religious beliefs oneducation. These studies cover a variety of areas such as religious education, cultural conflictbetween religious values and school culture, and impact of religious beliefs on teachers’identities. For instance, a recent book edited by Wong, Kristjánsson, and Dörnyei (2013)collates a series of research studies on the interrelationship of Christian faith and Englishlanguage teaching. As early as the 1990s, Cobern (1991) incorporated worldview (includingreligious beliefs) as an interpretative framework for the studies of science teaching andlearning. In a recent questionnaire study conducted by Aflalo (2013), it was found thatreligious beliefs had significant impact on preservice teachers’ beliefs about the nature ofscience, whereas their previous scientific knowledge and national culture only barely had animpact. The influence of creationism in Christian and Islamic faiths on students’ and teachers’conceptions of the nature of science in general and evolution in particular is one of the hotissues in science educational research. Some related literature includes publications byBoudJaoude et al. (2010), BouJaoude, Wiles, Asghar, and Alters (2011), Brem, Banney, andSchindel (2003), Clément et al. (2010), Mansour (2008), and Martin-Hansen (2008). In fact,the issue is not confined to particular religions such as Christianity or Islam. The studies aboveprovide evidence to support the contention that religious beliefs play an important role inshaping students’ and teachers’ beliefs about scientific teaching and learning.

Fewer studies have focused on mathematics education. In relation to students’ beliefs, Amirand Williams (1999) and Sharma (2006) found that religious belief is an influential factor onsome students’ probabilistic thinking. In respect of teachers’ beliefs, Norton (2002a, b, 2003)found that university mathematics teachers who subscribed to different religions showed verydifferent relationships between their religious beliefs and their mathematics research andteaching practices. The studies of Leu and her collaborators (Leu, 2005; Leu & Wu, 2004)revealed how mathematics teachers possessing Buddhist and Confucian beliefs might demon-strate subtle differences in how they view learning and teaching mathematics. More recently,Chan, Wong, and Leu (2012) and Leu, Chan, and Wong (2014) extended the scope of religiousimpact to all the popular religions in the Chinese regions (that is, Confucianism, Buddhism,Daoism, Catholicism, Protestantism) as well as those who do not subscribe to any religion. Itwas found that Catholic and Protestant respondents see mathematics more as a precise and“calculable” subject than their counterparts. Comparatively, respondents holding the traditionalChinese religious beliefs2 viewed mathematics as a school subject that involves thinking. Atthe same time, Catholic and Protestant respondents possessed relatively weaker constructivistviews on mathematics teaching than others. Subtle as they may be, the results revealed in theseinitial attempts were encouraging: religious beliefs do play a role in teachers’ mathematicsteaching (and consequently affect students’ mathematics learning). Though the methodologiesemployed have extended from questionnaire to open-ended questions in the above two studies,the major limitations were the absence of interactions between the respondents and theresearcher. Inevitably, more substantial data can be obtained if qualitative methods such assemi-structured interviews can be utilized. This is precisely the theme of the present paper.

2 Referring to Confucianism, Daoism, and Buddhism


3 Research question

Religious beliefs, as one of the major sources of one’s worldview, would naturally have anassociation with teachers’ beliefs about mathematics teaching and learning. However, theinterplay is so complex that the connection may not be a causal relationship. Furthermore,worldview is often deeply embedded in one’s mind and may not be so noticeable. Therefore, afirst step to unpacking the interplay is to investigate how mathematics teachers perceive theimpact of their religious beliefs on their own mathematics teaching and learning. This isexactly the focus of this paper. To be precise, we would like to investigate: how do mathe-matics teachers with different religious backgrounds perceive the connection between theirpersonal religious beliefs and their beliefs about mathematics teaching and learning?

4 Theoretical framework

Based on existing literatures on teachers’ beliefs about science education and his empiricalresearch study on Islamic science teachers’ views of relationship between science and theirown religion, Mansour (2008) developed a framework for understanding the interactionbetween individual teacher’s experiences, his or her own religious beliefs, pedagogical beliefs,and teaching practices. Mansour called this framework the PRB model. This model “beginswith the relationship between teachers’ personal religious beliefs, and their experiences andends with the relationship between teachers’ personal religious beliefs and their practices”(Mansour, 2008, p. 1,623). It highlights the distinction between “orthodox” religious beliefsand “personal” religious beliefs. There may be a wide gap between these two types of religiousbeliefs. It is the latter one which has significant impact on teachers’ pedagogical beliefs and(science) teaching practices. The notion of personal religious beliefs emphasizes one’s inter-pretation of life experiences in general and religious experiences in particular. This interpre-tation is built with the social context one lives in. Hence, this framework is a socioculturalapproach on understanding the relationship between beliefs and practices.

The study reported in this paper is about the relationship between teachers’ PRB and theirbeliefs about mathematics and mathematics teaching and learning. This study is somewhatsimilar to the study of Mansour (2008) in nature, namely that both focus on the relationshipbetween teachers’ religious beliefs and pedagogical beliefs. Furthermore, as pointed out byscholars such as Skott (2009) and Speer (2005), the relationship between (mathematics)teachers’ beliefs and practices is rather subtle and should be interpreted from a contextualpoint of view. As stated above, Mansour’s PRB model emphasizes the contextual factors in ateacher’s interpretation of life experiences and hence shaping of their pedagogical beliefs.Thus, it seems that this model as a theoretical framework is applicable to our current study(Fig. 3).

5 Methodology

Hong Kong, a meeting point of Eastern and Western cultures, is the context of investigation.Ninety-five percent of Hong Kong’s residents are Chinese. Buddhism, Daoism, andChristianity (including both Catholic and Protestant traditions) are the three largest religionsin Hong Kong. Although Confucianism is not as popular as Buddhism and Christianity, manyHong Kong residents (especially those in the older generation) are influenced by Confucianthoughts. There are also minority groups of Muslims, Hindus, Sikhs, and Jews. More details of


the cultural and religious background can be found in the publications by Leu et al. (2014) andWong and Tang (2012).

The objective of this research study was not to make inferences about a population on thebasis of empirical data collected. Rather, it purposed to investigate whether there is a possiblelink between teachers’ PRB and their beliefs about mathematics education. Thus, it focused onthe ways of linkage between these two types of beliefs as perceived by the participants.Therefore, purposeful sampling is adopted in order to find information-rich cases rather thanrepresentative cases (Patton, 2002). Fifteen mathematics teachers, teaching in primary orsecondary schools in Hong Kong, were invited to participate in the study. Among them, fivewere Buddhists, five were Christians, and five claimed not to subscribe to any religion. By wayof introduction, they were first requested to respond to a questionnaire which consisted ofseven open-ended questions on how they entered the profession, what kind of qualities theyexpect to nurture in their students, and how their religious beliefs might influence theirteaching. (See Appendix for the seven open-ended questions.) The question on how theyentered the profession (question 1) tries to trace how their worldviews (whether it is religious-related or not) might influence their decision to become a teacher. Questions relating to thekind of qualities teachers expect to nurture in their students (questions 2 to 4) try to reveal theirdeep beliefs (or in the terminology of Philipp (2007), their values) about education in generaland mathematics education in particular. The last set of questions (questions 5 to 7) focused ontheir perceptions of the relationship between religious beliefs and mathematics teaching. Thequestionnaire had two purposes. First, it enabled the participants to reflect on their beliefsabout mathematics education and possible relationship with their own religions. Second, itenabled the researchers to have an initial idea of the participants’ views on the issue. Theseinitial ideas were used as contextual examples to open up an individual semi-structuredinterview 1 week after the questionnaire was received. The individual interviews focused onthe participants’ views on mathematics education and religion. The interview questions weremodified from those by Cai, Perry, Wong, and Wang (2009), with additional questions about

Bolded arrows: Strong influence of one component on another, according to Mansour.

Teachers’ practices

Framework for action

Focus of this research study

Teachers’

social

contexts

Knowledge

Teachers’ worldviews and

personal religious beliefs

Teachers’ experiences

Teachers’ pedagogical beliefs

(particularly beliefs about

mathematics teaching and learning)

Teachers’

identity

Interpretation

Fig. 3 Personal religious beliefs model (adapted from Mansour (2008), p. 1,628, Fig. 3)


religious beliefs and their possible impact on mathematics teaching. The main interviewquestions were the following:

1. What is the essence of mathematics?2. What can a teacher do to help students learn mathematics well?3. What are the features of an effective mathematics lesson?4. Are there any similarities or differences between mathematics and religion?5. Does your religious belief have any impact on how you think about mathematics

education?

These questions tried to identify each individual participant’s beliefs about the natureof mathematics (question 1), beliefs about mathematics teaching and learning (questions2 and 3), worldview and how the worldview influences his or her beliefs about mathe-matics education and global ideology of teaching (questions 4 and 5). Content analysis ofthe interview was conducted. The interviews were audio-recorded and transcribed ver-batim. The transcripts were coded and coherence themes were generated. Mansour’s PRBmodel was used as an analytical framework. We tried to categorize different types ofPRB and different types of pedagogical beliefs (both mathematical and generic beliefs),and then identified how the interpretations of their own experiences influenced therelationship between these beliefs. Finally, the interview data were triangulated withthe questionnaire data in order to ensure the stability and reliability.

6 Results

In the following, three cases are presented. The participants in each case have different religiousbackgrounds: The first one is a Christian; the second one is also a Christian but is also deeplyinfluenced by Confucianism; and the third one is a Buddhist. The respondents have stated thatthey are devoted followers of their religions. All three cases show explicit links between their PRBand their teaching beliefs. They provide evidence that religious beliefs are not necessarilydetached from beliefs about mathematics and mathematics teaching and learning. Thus, thesethree cases can be regarded as “critical cases” (Flyvbjerg, 2006) of this study. It is only bycoincidence that all three teachers are female, and gender is not within the scope of this study. Theteachers’ names are pseudonyms.

The selection of quotations is based on the themes generated from content analysis of theinterviews. They are selected to illustrate how the teachers perceived the connections between theirPRB and their (mathematics) teaching beliefs. Narratives between quotations are included tohighlight the relationship between different themes. In the interview, the teachers also stated howtheir beliefs are presented through their own classroom practices. Readers are reminded that thedescriptions of these classroom practices are based on teachers’ self-reflective experiencesrather than researchers’ classroom observations. We are fully aware that the relation-ship between teachers’ beliefs and practices is a complex issue which involves socialcontexts (Mansour, 2008; Skott, 2009; Speer, 2005). However, we believe that the teachers’self-reported teaching experiences can illustrate how they think their beliefs can beenacted at least in ideal situations. How they actually enact these beliefs in theirclassroom settings is out of the scope of the current study, although this can be afollow-up study in future.

The interviews are conducted in Cantonese (the Chinese dialect of both the researchers andthe participants). The quotations are translated in English.


6.1 The case of Bonnie, a Christian teacher

Bonnie is a mathematics teacher teaching in a Christian primary school. She has taughtmathematics for 8 years and has been a Christian for 12 years.

6.1.1 Bonnie’s beliefs about the nature of mathematics: pattern seeking and logical thinking

Bonnie thinks that the essence of mathematics is the patterns hidden in our daily life.

I think that mathematics is a pattern. People discover some principles based on obser-vations, and then we apply them (the principles).

Therefore, she thinks that mathematics is a type of training of logical thinking.

The student learns mathematics, not for the purpose of competing with the calculator,[aiming to be] faster and more accurate than the calculator. No. S/he needs to learn thethinking skills. S/he can develop logical thinking skills [through mathematics].

6.1.2 Bonnie’ beliefs about mathematics teaching and learning: creating one’s own methodsand practicing routine methods are important

Bonnie emphasized the process of mathematical thinking rather than computation speed.Instead of memorizing routine procedures, she claimed that she likes to encourage her studentsto invent their own methods.

I like to ask [my students] these questions: How did you think last time? What methoddid you use? Why doesn’t this same method work this time? What is the difficult point?Based on the difficulty and the current problem, how can you improve your method?

Although Bonnie emphasized students’ invention of their own methods, she did not dismissthe importance of practice.

We also need to face the examinations. In order to improve the academic results within ashort period of time, certain types of questions such as computation questions requiremore practice.

However, Bonnie did not simply regard doing questions as drills for the sake of examina-tion scores. She pointed out a subtle relationship between practice and mathematical under-standing. She believes that it is important to practice even if the mathematical ideas are not yetfully understood by the students. She associated this belief with her own school learningexperience.

When I was at school, my mathematics teacher did not emphasize understanding. Hesimply asked us to do [exercises]. While simply following the teachers’ method, Igradually learnt how to solve the mathematical problems. But this may not be correct.For example, when I learnt differentiation and integration, the teacher did not explainmuch. I did not totally understand the method but just ‘created’ intermediate steps basedon the final answers [given in the textbook] to help me solve the problems. As mymethod worked every time, I continued to use the method although I did not totallyunderstand the mathematical concepts. Having matured and knowing more, I filled inthe gaps gradually. I understand the concepts a lot better. There may still be some gaps.But as I continue to work on it, the gaps can be filled in.


In brief, Bonnie emphasized mathematical understanding by means of both inventing one’sown methods and practicing on or following routine methods. She believes that these twomeans are not contradictory but complementary to each other.

6.1.3 How these beliefs were affected by her worldview: acquisition of both knowledgeand religious beliefs can have different paths and different paces

Bonnie made an interesting comparison between her beliefs about mathematics teaching andlearning and her religious experience. Ideally, a student should try his/her best to understandthe concepts and methods. However, it is practically difficult to understand them completely atthe start. She thinks that understanding could be enhanced bit by bit by means of practice evenif one simply follows the methods which one does not fully understand. She reflected on herown experience in developing her personal religious beliefs.

I think my path of looking for religious beliefs is similar to my view on mathematicsteaching. Some people think that one needs to understand the teaching of the Biblecompletely before becoming a Christian. For myself, I did not understand it completelywhen I took Christianity as my belief. I agreed with some basic things such as there isonly one God. Although I did not understand many [other] things written in the Bible, Ibecame a Christian. It (Bonnie’s view about religion) does not influence my view aboutteaching [directly] but there are commonalities between the two. In mathematics learn-ing, there are some basic things that you must be able to tackle but it does not matter ifsome details are missing. … [I]t does not matter if the students can only do some basicexercises but are unable to tackle the skills fluently. Keep going! They may understand itin future. I think it is similar to how one seeks for a religion.

Bonnie’s religious beliefs did not influence how she sees mathematics learning and teachingdirectly. However, as she pointed out, there is a certain degree of alignment between her beliefsabout religious growth and her beliefs about mathematics learning, which consequentlyresulted in emphasizing both students’ invention of their own methods and following routineprocedures to solve mathematical problems.

6.1.4 How Bonnie’s worldview influences her global ideology of teaching

Bonnie’s beliefs about mathematics teaching and learning seem to originate in her beliefsabout education in general, that there are different ways to learn something and that knowledgeis learnt bit by bit. As stated in the previous section, this belief about education is consistentwith how she sees the development of religious beliefs.

There is not just one path of learning. There may be many different paths. As a teacher, Ihope my students can understand [what I teach] first. But, I would not push (force) themto follow what I teach.…. They do not need to learn everything from me.… They (Thestudents) may not understand [what I teach] initially. Later on, a blurred concept maygradually develop, perhaps after some stimulation. …. However, the progress of [thestudents in] the whole class can be different. Some students understand what you (theteacher) say quickly and can build a learning path on their own, whereas some otherstudents may still be unable to handle [the most basic problem]. One needs to decide.Perhaps first practice (mathematics questions) according to what one already knows.

Bonnie’s remark appears not to be confined only to the learning of mathematics but also tohow she views learning in general. Nor are her views confined to academic subjects but also


other aspects of life including religious beliefs. It is difficult to say whether her religiousworldview influences her beliefs about education or vice versa. It may be that there is areciprocal relationship between her learning experiences (both at school and at church) andbeliefs about learning in particular and education in general.

6.1.5 Summary

Bonnie sees mathematics as a science of searching for patterns, which has been discussedthoroughly in curriculum documents such as Everybody Counts (National Research Council,1989). As pointed out in Everybody Counts, mathematics is more than just calculation ordeduction but also involves other methods and skills like observation, generation, and testingconjectures. It is indeed a recurring cycle between generation and application of rules andprinciples derived from patterns. At a pedagogical level, Bonnie de-emphasizes calculationspeed and accuracy which can be done by calculators. Rather, she emphasizes developingstudents’ mathematical thinking skills, just like the notion of “re-invention” of Freudenthal(1991) that “Children should repeat the learning process of mankind, not as it factually tookplace but rather as it would have done if people in the past had known a bit more of what weknow now” (p. 48). She claimed that she encourages her students to invent their own methodsor even “invent” the intermediate steps based on the final answer of the question. On the otherhand, with the reality of examination requirements, she did not dismiss the importance ofroutine practice. However, her emphasis on practice is different from drilling for the sake ofexamination scores. Indeed, she thinks that it is important to practice even if the mathematicalideas are not yet fully understood because practice is an important way to facilitate mathemat-ical understanding. This suggests a reciprocal relationship between practice and understanding,which is consistent with earlier findings on experienced mathematics teachers’ perspectives oneffective teaching (Wong, 2007; Cai et al., 2009; Sun, Wong, & Lam, 2005).

Bonnie compared her view on the relationship between practice and mathematical understand-ing with her personal experience in searching for a religion. She pointed out that the reciprocalrelationship between practice and understanding is not restricted only to mathematics learning butalso applies to religious growth. One starts to follow a religion with some basic (but incomplete)understanding. When one puts religious faith into practice, one can arrive at a deeper understand-ing, which would lead to more commitment to one’s religious beliefs and further put these beliefsinto actual practice. Indeed, the interaction between faith and reasoning is an emphasis inChristianity. Theological reflection (or more generally, thinking about any issue from aChristian perspective) is usually based on multiple “resources” including Scripture, tradition,reason, and experience (Stone&Duke, 1996/2006). Bonnie applied this spirit of thinking not onlyto her religion but also to her teaching.

In a broader sense, Bonnie’s beliefs about the reciprocal relationship between practice andmathematical understanding reflect her beliefs about education in general, that learningsomething (whether an academic subject or religious knowledge) is a bit by bit process inwhich different people may have different paths and paces. It matches well with the ideologyof appreciation for diversity (Habermas, 1993) which is derived from Christian theology ofCreationism. An implication for (mathematics) education is that some people have greaterability to study mathematics, whereas others may possess abilities to study other academicsubjects or have talents to do other nonacademic things. Each student has his or her owncapacity and learning pace. A teacher should respect the individual differences of his or herstudents. In particular, she emphasized both students’ invention of their own methods andfollowing routine procedures, which is an appreciation for diversity in her students’(mathematics) learning methods. Figure 4 shows the belief clusters of Bonnie.


6.2 The case of Carol, a Christian who is also influenced by Confucianism

Carol is a mathematics teacher teaching in a Christian secondary school. She has taughtmathematics for 30 years and has been a Christian for 20 years. At the same time, she statesthat she is deeply influenced by Chinese culture and Confucianism. She began learning aboutChinese culture and reading Chinese classics3 when she was a Grade 4 student.

6.2.1 Carol’s beliefs about the nature of mathematics: mathematics has absolute rightand wrong

Carol holds an absolutist (infallible) view of mathematics (Lakatos, 1976). She considersmathematics to be very clear-cut and to have an absolute right or wrong independent of humansubjective opinions.

Mathematics is either black or white; either right or wrong. It cannot be just a little bitcorrect. It cannot be neutral. There is no such thing in mathematics. Mathematics is veryclear-cut. If your answer is correct, I must say that you are correct. Even if I dislike you, Icannot argue that your answer is wrong if it is indeed correct. If your answer is wrong, Icannot say that you are correct no matter how much I like you.

6.2.2 Carol’s beliefs about mathematics teaching and learning: logical reasoning is the coreof mathematics education, which is not only important in the subject disciplinebut also a means to teach moral values

Carol’s absolutist view of the nature of mathematics is consistent with her belief about howmathematics should be learnt. According to this perspective, the only valid warrant ofmathematical claim is logic and deductive reasoning. Basically, she believes that this is theonly way of doing mathematics. Therefore, in her opinion, logical reasoning is of utmost

3 Carol did not explain what Chinese classics she read. However, we inferred from our interview that she wasreferring to the Confucian Canon of Four Books.

Personal learning

experience

Seek the religious

beliefs bit-by-bit

Learning can have many

different paths

Do practice even if the

mathematical content is not

totally understood

Mathematics is pattern seeking

and logical thinking

Emphasize the mathematical

thinking process

Encourage students to

construct and reflect their

own problem solving

methods

Fig. 4 Bonnie’s belief clusters and their connections


importance in mathematics learning. In her view, logical thinking ability is a necessaryrequirement for learning mathematics.

It is impossible to learn mathematics if one does not think. … One needs a correctlogical inference: first, the premise; then, inference; and finally, the conclusion. This isthe essence of mathematics, especially in deductive geometry.

Furthermore, she believes that learning logic and deductive reasoning as well as thethinking principles behind mathematics is the basic goal of mathematics education.

Why do you need to learn mathematics, especially since we have calculators? You donot really need to learn computations because you have calculators. … The method ofmathematical thinking is essential to our daily life. You are not teaching students to docalculations but to teach them logical thinking, methods of inference and philosophy ofhuman life.

Notably, Carol believes that mathematical thinking methods are not only a useful tool forsolving mathematical problems but also, philosophically, a guiding principle of how we shouldbehave in daily life. She elaborated,

You should not turn black into white, turn curve into straight. Curves are curves; straightis straight; black is black; white is white! However, this may not be the case in real life.In real life, it is difficult to distinguish between good and evil; but it is the case inmathematics. If [the size of] an angle is 30°, it is 30°. You cannot say that it is 31°. Weneed to learn this moral value. When I teach deductive geometry, I teach this philosophyof human life. We teach students correct moral values through mathematics.

In other words, Carol believes that mathematics is a good platform for transmitting moralvalues such as righteousness and distinguishing between right and wrong. As an illustration,she described how moral values could be touched upon through the topic of solving equations.

You set an equation. Why did you set it in this way? Is it just a guess at the answer?Nowadays, it appears acceptable that one earns money, no matter whether it (the money)comes from gambling or cheating. Making money is a good thing, but even bad peoplethink in this way! In this, the target is to earn money, no matter what method has beenused. In relation to mathematics, our students try to make up the answers by any method.… In mathematics, we need to know why we get the answer. Where does the answercome from? Why do we need to solve it in this way?

6.2.3 How these beliefs were affected by her worldview: her philosophy of human lifeis the Confucian thought of “doing right things”

Carol’s opinion about teaching moral values through mathematics is consistent with herpersonal philosophy of human life. She claimed that she is a person who aims to “do rightthings”. She made an interesting analog between how one should behave and how to domathematics questions.

My philosophy of human life? Doing the right things. If you do one wrong thing, thesteps that follow will also be incorrect. For example, if I steal money, this is [morally]incorrect, and I will also be wrong to use this money. Applying this to solvingmathematical problems, if my first step is incorrect, my second and third [steps] willalso be incorrect.


Carol pointed out that her philosophy of human life developed since childhood andoriginates from traditional Chinese Confucian thoughts.

It [Emphasizing ‘doing right things’] is moral standards and Confucian thoughts. I learntmoral values and Confucianism when I was in Grade 4. For instance, obeying ourparents is [implied in] the five cardinal human relations.4 These are only basicmoral standards. I learnt these Chinese values when I was young. I follow thesevalues which do not contradict my religion. No matter which religion webelieve, we need to follow these moral standards…. [They] are taught in theConfucian and Daoist classics. All who studied these classics follow these norms nomatter what religion we subscribe to.

6.2.4 How Carol’s worldview influences her global ideology of teaching

Carol’s philosophy of human life (“doing right things”) which originates in Chinese culture andConfucian thoughts greatly impacts her beliefs about education in general and mathematicseducation in particular. Apart from Chinese cultures and Confucian thoughts, she claimedthat she has been a Christian for 20 years. It is quite reasonable to expect thatChristian religious beliefs should also have some contributions to her worldview andideology of teaching. In response to the possible impact of Christianity on her beliefs aboutteaching, she said,

God gave me the chance to take up this career. I do it seriously. The impact [of my religion]is that I should work hard with whatever God gives me [and asks me to do]. No matterwhich religion I believe, I should be responsible. Even if I do not believe in Christianity ordo not subscribe to any religion, I have to be responsible [to my job].... I have a mission. Ibelieve in Jesus. I have a grateful heart.... I believe that I am just a humble woman. I cannotdo anything by my own effort… I work hard at whatever God asks me to do.

She concluded,

Relating to my philosophy of human life, I have a strong sense of right and wrong whichcomes from Confucian thoughts. This is my framework [of human life]. What [Chris-tianity] enriches me is God’s grace. My religion gives me so much grace. I know that Iam just a humble woman. God put me in this good environment. I need to tend His[God’s] kids, the little sheep. I need to work hard.

Carol’s religious beliefs reinforce her philosophy of human life that “right and wrong”judgment and “doing right things” should be emphasized. She believes that working hard atteaching is the right thing to do. Her religious beliefs give her a spiritual reason (namely, God’sgrace and mission) to put her philosophy of human life into her teaching practice.

6.2.5 Summary

Carol holds an infallible view on mathematics. She thinks that mathematics is infallibleknowledge, and hence mathematical truth is clear-cut, unambiguous, and is secured by logicalstructure. From this perspective, the methodology of mathematics is deductivist, which is best

4 Five cardinal human relations (wu lun) is an important tenet in Confucianism. It “refers to the five dyadicrelationships between ruler and minister, father and son, husband and wife, older and younger brother, andfriends” (Sun, 2013, pp. 11–12).


demonstrated in Euclid’s Elements. It begins with definitions and axioms, and then followswith lemmas, theorems, and proofs. This conception of mathematics, which is usually knownas absolutism, has a long history which dates back to Plato. Yet, there are many otheralternative views about the nature of mathematical knowledge and mathematical truth whichare more widely accepted today (Tymoczko, 1998).

Despite the fact that the absolutist view is not so widely accepted nowadays, Carol extendsthis perspective to the “moral aspect” of mathematics. Because of the clear-cut and absolutenature of mathematics, she believes that mathematics is a good platform for teaching moralvalues. For instance, righteousness can be taught to students through mathematical topics suchas deductive geometry or even through simple equation solving. The power of mathematics inshaping one’s character was an important theme in classical texts. The contribution ofmathematics to the training of the soul in general and moral values in particular is emphasizedin Plato’s works. In his Commentary on the First Book of Euclid’s Elements, Proclusinterpreted Plato’s thought and pointed out the dianoetic intellection of mathematics throughits beauty and order (White, 2006). This view echoes how ancient Chinese intellectuals seethe values of Western mathematics. When Euclid’s Elements was introduced to ancientChina during the Ming dynasty, the translator Xu Guang-qi, who was brought up inthe Confucian tradition but later became a Catholic Christian, wrote the followingcomment on Elements in his article entitled Ji He Yuan Ben Za Yi (Various Reflections on JiHe Yuan Ben5):

Five categories of personality [one] will not learn from this book [Euclid’s Elements]:those who are impetuous, those who are thoughtless, those who are complacent, thosewho are envious, and those who are arrogant. Thus to learn from this book one not onlystrengthens one’s intellectual capacity but also builds a moral base. (As translated bySiu, 2008, p. 360)

Indeed, Carol’s perspective appears to be a “duplication” of the story of XuGuang-qi: not onlydid she think that deductive geometry (a mathematical methodology and style of presentation asdemonstrated in Euclid’s Elements) can build the moral base of her students but also that Carolhas a similar background to Xu Guang-qi—brought up in the Confucian tradition and laterbecoming a Christian. Indeed, she attributed her perspective on the connection between mathe-matics and moral education to Confucian thoughts which nurtured her since childhood.Confucianism emphasizes rite and role. Everything has a proper way and each person shouldbehave according to his or her role. One should take up one’s responsibility according to one’sown role, and try one’s best to do so. The emphasis of “rite”, i.e., everything has a method, is acentral notion of Confucianism. That is where “doing the right things” comes from. InConfucianism, the belief that everybody is educable is prominent (Lee, 1996). Students shouldstudy hard to “repay” the blessings of their parents. Teachers take heart in teaching not just fornurturing their students but to continue the (cultural) heritage which was laid upon their shoulders(Wong et al., 2012).

Apart from being brought up in the Confucian tradition, Carol also has been aChristian for quite a long period of time. The “interaction” between Confucianthoughts and Christian beliefs (and its influence on mathematics education) is rathersubtle. “Doing right things” is not only emphasized in Confucianism but is also acore teaching in Christianity. Following the Law of God, which includes a moral aspect,

5 Ji He Yuan Ben is the Chinese title of the Elements translated by Xu Guang-qi.


is a repeated theme in the Bible. However, Carol did not attribute “doing right things” toChristianity but Confucianism. In the interview, she emphasized that it is not related to herreligion but to Chinese values and moral standards that one should follow no matter whichreligion one believes. One possible reason is that Confucianism does not touch upon theism anddoes not care too much about the afterlife. In other words, in the eyes of Carol,Confucianism and Christianity have very different concerns. It is in the generic (non-mathematics specific) aspect that Christianity has an impact on Carol’s beliefs about education.Carol sees teaching as a grace and mission given by God. She used the metaphor of “tend his[God’s] kids, the little sheep” to describe the relationship with her students. The “shepherd-and-sheep relationship” is widely used in Christianity to describe the relationship between Jesus andChristians as well as the relationship between pastors and churchgoers. Carol extended thismetaphor to the relationship with her students. It suggests that she regarded herself not only as ateacher but also as a church pastor. This metaphor suggests a dual function in the relationship.On the one hand, it suggests the function of (spiritual) caring; on the other hand, it suggests thefunction of teaching of religious values (including moral values). To a certain extent, althoughCarol may not be aware, it appears that Christianity provides her with a religious reason to puther philosophy of human life such as “doing right things” into actual practice in everydayteaching. Figure 5 shows the belief clusters of Carol.

6.3 The case of Apple, a Buddhist teacher

Apple is a mathematics teacher teaching in a Buddhist secondary school. She has taughtmathematics for 18 years and has been a Buddhist for 3–5 years. She reflected that she was notsure when she began her search for Buddhist teaching. She believed that it may be as early aswhen she started her teaching career.

6.3.1 Apple’s beliefs about the nature of mathematics: a stable system of knowledge

Apple sees mathematics as a logical discipline and a system of knowledge that is independentof human culture. This is different from seeing mathematics as an evolving humancultural endeavor. Her belief about mathematics resembles an absolutist (infallible)view.

Mathematics is a scientific language, a tool and…it is a knowledge; a knowledge oflogic seeking. It is problem solving by logic. [It is] just something independent of humanbeings. [That’s why] it is fun.

6.3.2 Apple’s beliefs about mathematics teaching: connection should be emphasized

Apple emphasizes the structure of mathematical knowledge. She looks highly upon theconnection among different mathematical topics and thinks that students should see mathe-matics from a wider perspective.

S/he (the student) needs to tackle mathematics as a whole. The examples and problemswhich may appear different are actually [embedded in] one concept. S/he needs to seethe connection. When s/he sees the connection, s/he can handle the concept [as a whole]and can see the key to solving problems. Some students cannot do mathematicsproblems that vary slightly but involve the same concept because [they] cannot


follow the line of thought. In order to see the connection, one needs to have awider perspective.

She further explained what she meant by having a “wider perspective” in relation to thesenior secondary school mathematics syllabus.

Just like in A-level6 [syllabus], we have the topics ‘Coordinate Geometry’, ‘Vectors’ and‘Complex Numbers’. In fact, when you study them together, you will find that thesethree topics are actually ‘one piece’. There are many commonalities within.

6 Advanced Level

‘Doing right things’ as her

personal human philosophy

(rooted in Confucianism)

Teaching as a

religious mission in

Christianity

Emphasis on teaching students to distinguish between

(morally) right and wrong

Mathematics is logical inference and has

absolute right-or-wrong judgment

Emphasis on learning the method of logical

thinking in mathematics

Teach moral values through mathematics teaching

Fig. 5 Carol’s belief clusters and their connections


6.3.3 How these beliefs were affected by her worldview: everything is within the Buddhistdoctrine

Apple emphasizes seeing from a wider perspective and looking for the connection amongdifferent mathematical topics. This is not confined to mathematics but also applies to acqui-sition of knowledge in general, including her personal religious beliefs. In fact, she maintains awide perspective in Buddhist doctrine, which is not uncommon in Buddhism.

Everything is [within] Buddhist doctrine. Buddhist doctrine is about [the law of] Heavenand Earth. You should realise that Buddhist doctrine is purely a label. Underlying it arethe patterns of nature [including] those insights that you can perceive and those insightsthat you cannot. For instance, physics is the law of objects. Buddhist doctrine is aboutthe events in our daily life. All things and events, like interpersonal relationship and hotnews, have their underlying laws. We cannot see the laws explicitly but through all thesethings [phenomena]. Mathematics is a theory of something. It is a way to understand thepatterns of the real world through our human mind. This is the ‘way’ and [within]Buddhist doctrine. Therefore, in order to understand something, you need to see thingsfrom a wider perspective.

Apple sees knowledge about the world as a whole. On the one hand, mathematics has its owncoherent system which connects different mathematical topics. On the other hand, mathematicsrelates to the “patterns of realistic world” and hence is closely connected with other disciplines.Moreover, Apple put the knowledge about the real world under the umbrella of Buddhistdoctrine. One should realize that all knowledge is actually “[the law of] Heaven andEarth” and ultimately points toward her own religious belief of the Buddhist “way”.This perspective of integrating religious belief with (secular) knowledge is rather differentfrom what one may usually perceive.

6.3.4 How Apple’s worldview influences her global ideology of teaching

Buddhist doctrine does not only influence how Apple sees the acquisition of knowledge butalso her attitudes toward her students and her (nonsubject specific) educational goal. First ofall, she pointed out that the Buddhist doctrine of “cyclic existence”7 changed her way of seeinglife. At the beginning, this doctrine led her to have a pessimistic view of life. Through theprocess of spiritual practice, her view of life has become more optimistic.

First of all, I have a better way of seeing life. I did not simply focus on this very [orcurrent] life. When I first knew [of the Buddhist teaching] that life lasts forever, I feltvery horrible. I imagined life like a never-ending road. If [you walk] happily forever, it isOK; but sometimes, people live painfully, then forever is terrible as one cannot seeliberation. In the process of spiritual practice, I achieved the [inner] bliss gradually. Iinterpret [the doctrine of] ‘life is forever’ differently. What you are facing at thismoment, no matter good or bad, will pass away. You cannot guarantee what will happenin future but you have to keep being positive. Life is forever, passing from one life to thenext. Just keep a bright and transcendent mind.

7 “Cyclic existence” (or samsara) is one of the core Buddhist doctrines. It means that one continues to be rebornin the various forms of existence, no matter whether human, animal, or as other beings.


She further explained how this new interpretation of Buddhist doctrine led her to have anew way of seeing her students.

It has an implication for how I see my students. I have greater sympathy. I can see theirhopelessness and the dark side of their minds. They are ruled over by negative forces[thinking] such as lack of motivation to work. I should take care of them rather than justsaying that they are useless. They [The students who are not academically strong] havetheir own difficulties. I help them bit by bit—try to help them to walk on the brighter side.

In order to help her students “walk on the brighter side”, she claimed that mathematics can be atool to develop personal growth such as getting back the confidence in and having the joy oflearning.

Mathematics is a tool. They (the students) get a sense of satisfaction through learningmathematics. Each day, when they enter the classroom, they learn something and areappreciated by the teachers. They develop their value [self-worth]. …When they findthat they have learnt something and can do the [mathematics] questions, they would behappier. Well, some of my students are really very weak [academically]. I do not knowwhether they can pass the DSE8 after 6 years [of study]. Let them enjoy every lesson inthese 6 years, they would have a happy life which is very important to their souls.

In brief, Apple believes that an implication of the doctrine of “cyclic existence” is that oneshould live happily and keep a bright and transcendent mind no matter whether the currentsituation is good or bad. This religious worldview appears to apply not only to her life style butalso to how she interacts with students. Through mathematics teaching, Apple aims to guideher students to be free from “negative forces” and move forward to a bright and positive life.

6.3.5 Summary

Apple emphasizes coherence and connection not only within mathematics but seeing mathe-matics as a way to understand the patterns of the real world and more broadly the Buddhist“way” because she believes that everything is related to and embedded in Buddhism. Hence,both Buddhist doctrines and the theory of real-world objects (mathematics, in particular) guideher to understand “the patterns of the real world”. This connective epistemological worldviewis not uncommon among Buddhists. Apart from connecting mathematics learning to theBuddhist “way”, the notion of cyclic existence also has an influence on Apple’s genericeducational goal. Cyclic existence is one of the core notions of Buddhism. Buddhists’ attentionis not confined to this very (or current) life. Inspired by this notion, she believes that it is ofutmost importance to lead her students to be free of negative forces and move with a bright andtranscendent mind. The path to enlightenment, which may be taken to be synonymous withbeing fully “educated”, is a long course of numerous lives. Being foolish (or academicallychallenged) in this life does not mean that one cannot be more intelligent in the next. Thisgives a great hope to education. Everybody possesses the seed to enlightenment, i.e., everyoneis educable. Whether one is successful depends on one’s own effort (rather than on theblessings or curses of a godhead). The teacher (or spiritual guide) serves to provide the studentsoil for growth (Wong et al., 2012). Figure 6 shows the belief clusters of Apple.

8 Diploma of Secondary Education examination, taken at the end of 6 years of secondary school study.


7 Discussion

In this section, we compare the three cases to derive a fuller “picture” of the issue. Implicationsfor methodological issues and further research directions as well as pedagogical implicationsare then discussed.

7.1 Revisiting the three cases

It is apparent from the analyses that, no matter how subtle, there is evidence that teachers’beliefs about mathematics teaching and learning align with their personal religious beliefs.What is different from other studies is that these teachers were not identified by the researchersas followers of certain religions but that they had declared their religion themselves.

Mathematics is a logical

system of knowledge which is

independent of human beings

Everything is within

Buddhist doctrines

In order to understand

something, one needs to see

things from a wider perspective

Emphasize connection between

different mathematical topics

Seeing mathematics from a wider

perspective

Life as a process of cyclic

existence

A teacher should guide

her students to escape

from negative forces and

move to a brighter life

Mathematics as a means for

students to get back the

worth and joy of learning

Apples’ Buddhist beliefs

Fig. 6 Apple’s belief clusters and their connections


In the first case, Bonnie’s teaching allows for her students to understand mathematicalknowledge bit by bit, looping around between concept formation and computation. On the onehand, this is consistent with what Pierie and Kieren (1989, 1992), Sfard (1991), and Gray andTall (1994) revealed: procedural and conceptual understandings are not segregated andunderstanding happens in a “zig-zag” way. On the other hand, Bonnie mentioned explicitlythat such an approach was similar to her own religious experience: first believe by faith, thenexperience the grace of God as one goes along. Turning to Apple’s case, she sees that both theobject of learning and the course of learning are unlimited, in line with her Buddhist views thatboth space and time are boundless. The phenomenal world is not restricted to the earthlyregion we experience and conceive of. At the same time, the course of living is not confined tothis life (Wong et al., 2012). Different pieces of knowledge are connected. There could bevastly different paths (of learning and nurturing) and one should not get frustrated if onetemporarily finds difficulties in moving along a certain path. Finally, in Carol’s case, shereflected that she is influenced by both Christianity and Confucianism. Clearly, these tworeligions have fundamental differences in both ideology and in practice. However, it has beenargued that Confucianism is not a religion since it does not involve theism (Confucius is not a“god”) and there is, as such, no apparent conflict between the two (Wong et al., 2012).Nevertheless, what Carol has applied from her Christian beliefs is that teaching is a far greatermission than simply the delivery of knowledge. Spiritual growth is one. In relation to herConfucian beliefs, she holds that everything has a certain “right way” (Wong, 2004). Thesetwo notions are in fact rather common in Christianity and Confucianism (or even some otherreligions). As the writer of Proverbs in the Holy Bible has said, “Train up a child in the way heshould go: and when he is old, he will not depart from it” (Proverbs 22:6).

When we put these three cases together, we see threads of connections between personalreligious beliefs and beliefs about mathematics teaching. What is more interesting is thatvarious religious teachings were not directly “translated” into teaching. Rather, differentteachers, probably affected by their own experiences, took different parts (or ingredients) intheir religions and internalized them into their teaching beliefs. These ingredients are portrayedin Fig. 7. Their worldviews, as revealed above, in turn influence not only teachers’ beliefsabout education in general but their beliefs about mathematics teaching. This observation isconsistent with the reciprocal relationship among teachers’ PRB, experiences, and pedagogicalbeliefs, as stated in Mansour’s PRB model (Fig. 3). Inevitably, for the ordinary person, evenfor those who possess religious faith, their understanding of and devotion to a particularreligion may not be that apparent, so their teaching beliefs may not exhibit their religiousbackground too much. In relation to our respondents, the linkages between religious beliefsand teaching beliefs were more prominent.

Our general observations and interpretations are as follows. Any religion constitutes a vastsystem, from how to see the world, to how to behave, and how to build a spiritual life (or arelationship with God). Social issues (like education) may not occupy a core part in religions.For instance, Buddhism is about getting enlightened, and so gender issues may not be its majorconcern. (After all, to a Buddhist, a male can be reborn many times as males or females in theircountless cycle of rebirth and, in that sense, it is difficult to consider an individual as male orfemale.) Likewise, the enhancement of discursive thinking is not the goal of Confucianism(Wong et al., 2012). Thus, without revealing whether any teacher would refer loyally to theirreligious faith in all their actions, it is rather impossible for most followers to comprehendthoroughly every corner of such a huge religious system. A more realistic picture is that anindividual appears to take only some pieces from his/her religion and uses these as guiding postsin his/her (teaching) behavior. As noted from Carol’s case, one can take reference points frommore than one religious belief. Comparing Carol’s case to Bonnie’s, two individuals influenced


by the same religion can pick out different points for reference. In Apple’s case, she believedthat both the object of learning and the course of learning are unlimited, and hence mathematicslearning is a “way” to understand the phenomenal world—not just this earthly world but alsothe spiritual world. This perspective has a resemblance to the Christian perspective that there isno distinction between religious knowledge and secular knowledge. All knowledge comes fromGod, and a human’s ability to comprehend knowledge is limited and incomplete. As animplication for school education, Christian teachers should investigate the possible connection

Christianity * Confucianism

Believe by faith first

then experience the

grace of God

Guide your kids

etc #

etc

Everything is within

the Buddhism doctrine

Life is cyclic

existence

Equity (^)

Educability of

all(?) love (?)

etc

Everything has a right

way of doing

Buddhism

Apple’s worldviews

Carol’s worldviewsBonnie’s worldviews

¥

etc

* We do not argue whether these points are orthodox or layman

# We do not aim to provide an exhaustive list which is out of the scope of our study.

¥ Obviously, these are not the only sources shaping their worldview

^ We state only some points (e.g. love, equity …) as examples. We fully realize that all these features,

though bearing the same label, have subtler fundamental differences in different religions.

---: threads of connections

Fig. 7 Religious origins of worldviews


between school subjects and their Christian faith. Furthermore, Apple believed that the mostimportant thing in education is to keep a bright and transcendent mind. In a nonreligiousspecific way, it can be inferred that one should always keep a joyful mind and a positive way ofthinking. Although the rationale of religious doctrine is very different, keeping a joyful mindand a positive way of thinking as one’s life style is also emphasized in many other religions likeChristianity. However, Apple, being a Buddhist, associates these values with Buddhist doc-trines. Would she associate them with Christian doctrines if she were a Christian?

Another interesting observation is that all three teachers mentioned that either their learningexperiences or teaching experiences play a role in shaping their beliefs about mathematicseducation, which is in turn “reinforced” by their own religious beliefs. This point supports ourclaim at the beginning of this article that religious beliefs are so rich that any teacher may takeonly a portion of them to inform his or her (mathematics) teaching, which manifests one’ spersonal religious beliefs. As pointed out by Mansour (2008), personal religious beliefs (andtheir impact on teaching) are influenced by many factors such as learning/teaching experience(life history), classroom context (sociocultural context), and one’s own interpretation ofreligious beliefs. It is the teacher’s personal religious belief rather than the whole religioussystem that matters. It leads to the issues of methodology and further research direction, whichare discussed in the next section.

7.2 Issues of methodology and further research direction

It is an over simplification to assume the existence of some common religious values shared byteachers with the same religion. As mentioned above, it may be that the beliefs about(mathematics) education of two teachers subscribing to the same religion are influenced bytwo very different religious values. It may even be the case that a teacher is influenced by thevalues of a number of religions (and of course some other values which do not originate inreligions as well). To have a more precise picture of such a linkage, we have to shift fromconsidering “religions” as a unit of analysis to “religious/spiritual values” (see bullet points inFig. 7). The first step may be the identification of such values within traditional religions. Thiscan be done by extensive documentary analyses of scriptures involved and conducting large-scale case studies. Methodologically, incorporating open-ended questions like “my religionis…” (refer toWong, 1993, for details on such questions) can help in extracting such values andgradually sorting them out into a list of religious/spiritual values. We believe that with suchgroundwork at hand, we can reveal a tighter linkage between pedagogical beliefs and religiousvalues (instead of religions per se). The present study can already be extended to other religions,but with the above list of religious/spiritual values at hand, it can be easily extended to thosewho claim not to subscribe to any religion. (One can be spiritual even though one is not afollower of any religion.) This group of people is in fact quite mixed in nature.

Sampling is another methodological issue that needs to be considered. Although we did nothave such an intention, all three participants happen to be experienced teachers and devoutfollowers of their own religions. It is presumed that an experienced teacher would have gonebeyond the “survival stage” in their teaching career and developed relatively more stable beliefsabout mathematics education. Hence, it should be relatively easier for them to reflect on howtheir personal religious beliefs may impact on their beliefs about mathematics education. Thismay explain why the three participants could provide such rich data in a short interview. On theother hand, it is equally worthwhile to investigate new teachers who are also devout religiousfollowers. As a new teacher may still be at the “survival stage”, their beliefs about mathematicseducation may still be developing. Investigating how one’s personal religious beliefs shapeone’s beliefs about mathematics education should provide us with some useful insights on how


religions may influence teachers’ beliefs about (mathematics) education. A longitudinal studyon new mathematics teachers’ changes in beliefs about mathematics education and the role thatreligions may play in such changes can be a possible research agenda.

In order to ensure that one’s personal religious beliefs have real impact on the personalworldview, it is important to recruit participants who are devout religious followers. Althoughthe number of years of subscribing to a particular religion is an indicator, it may not be asufficiently accurate way to identify whether a potential participant is a devout follower or not.A validated instrument could be developed to fill this insufficiency. To date, there are quite anumber of well-established instruments on religious engagement which are based on Westernreligions (Hill, 2005). There is a need to develop such instruments to include Eastern religionsand those who claim to belong to no religion. As mentioned earlier, the list of religious/spiritual values can be a basis for developing such instruments.

Last but not least, this paper describes the case study of three mathematics teachers’ self-reported beliefs about mathematics teaching and learning and their relationship with religiousbeliefs. It is the first step to investigate how teacher’s personal religious beliefs may influencetheir beliefs about mathematics education. It is a long road ahead for investigating how thesebeliefs may cause actual impacts on teachers’ classroom practices and consequently students’learning outcomes (Fig. 2). In particular, there may be inconsistencies between (mathematics)teachers’ beliefs about mathematics teaching and their actual teaching practices (Furinghetti,1997; Wen & Leu, 2004). Teachers may have some unconscious beliefs, like a ghost,influencing their teaching behaviors (Furinghetti, 1997). Furthermore, whether the teacherspossess a strong will is also a crucial factor leading to the consistencies between teachingbeliefs and teaching behaviors (Leatham, 2006; Wen & Leu, 2004). To proceed further, apartfrom relying on teachers’ self-reported teaching and learning beliefs, one can conduct class-room observations to check against the degree of consistency between their beliefs and theirteaching practices. Subsequently, a post-lesson follow-up interview can be conducted to revealthe teachers’ rationale for their teaching decisions and how their beliefs, both personalreligious beliefs and beliefs about mathematics teaching and learning, may influence theirdecisions (Skott, 2009; Speer, 2005). We believe that our first step here provides a foundationfor such further studies.

7.3 Pedagogical implications

We now turn to the discussion of pedagogical implications. It is now widely accepted thatmathematics is not culture-free and value-free (Bishop, 1988). Mathematics in itself is a humancultural endeavor (Wilder, 1952). Religion is yet another major human cultural activity. Asmathematics and religion coexist in human culture, there are bound to be interactions betweenthe two, nomatter how strong or subtle theymay be. The results of the present study do reveal sucha linkage. When we come to pedagogy, the design of learning activities in the mathematics classneeds to consider the religious/cultural setting of the classroom (both on the teachers’ and students’sides). This issue would be particularly salient when we have a multicultural society where peoplewith different religious backgrounds live and study together (Biggs, 1999). In particular, it may beimportant to take into account the cultural needs of religious minority groups when designinglearning activities, even in the seemingly culturally “neutral” subject of mathematics.

The above may be equally true for educational ideologies. We have heard from ourrespondents that “mathematics is a science of patterns”, “re-invention”, and “equity”. Ourrespondents may not have had the chance to read through literature such as Freudenthal’sworks or Everybody Counts. But these notions have become so popular and well received thatthey have appeared on curriculum documents. We may even term it as a “curriculum


worldview”. However, the generation of “curriculum worldview” is not automatic. Forinstance, the democratization of education was put forth in the 1910s (Dewey, 1916; seealso, Stemhagen & Smith, 2008) under the bigger concern of (political) democracy. Therecould be regional (for example, east-west) differences too. The social, political, and eveneconomic and cultural/religious backgrounds of a period generate a social mood that affectsthe curriculum worldview, which in turn affects the teachers who are delivering the curriculumas well as the students who are “consuming” the curriculum. For instance, under the samewave of democracy of education, more should be done to allow students to take ownership oftheir learning. In recent years, the issue of student and teacher centeredness has raisednumerous discussions. It is not the scope of this study to go into these. However, we simplywanted to highlight that teachers, who are delivering the curriculum, can have deeperreflections. In fact, in recent years, notions like learning centeredness (instead of learnercenteredness: Watkins, 2008) have been put forth and the possibility of “teacher-led yetstudent-centred” learning has been raised (Wong, 2009). All in all, since we should not takefor granted that there is a set of universal cultural/religious worldviews, the consequentialcurriculum worldview is not unique. When we are implementing a curriculum developedunder a worldview, we can also reflect if there are other approaches to teaching.

7.4 A final remark

As a final remark, in a modern multicultural society like Hong Kong, people may not haveobviously segregated religious beliefs. It is possible that they are influenced by a number ofreligions. Like the case of Carol, though she has been a Christian for 20 years, she attributesher emphasis on “doing right things” to Confucian thoughts rather than rooted in her ownreligion. This example highlights the complicated relationship between philosophy of life andreligious beliefs (and consequently, beliefs about mathematics teaching). Religious belief isdefinitely a source of one’s philosophy of life. However, it may not be the only source ornot even a major source. Despite the difficulties in separating a teacher’s religiousbeliefs and his/her philosophy of life, the (potential) impact of teachers’ religiousbeliefs and philosophy of life on the beliefs about mathematics teaching is an important yetunderdeveloped research area.

Acknowledgments The authors would like to thank all teachers who contributed to this study. We are gratefulto Prof. Chi-Chung Lam and Prof. Corinne Maxwell-Reid for commenting on an early draft of the manuscript.Special thanks also to the three anonymous reviewers and the editor Dr. Norma Presmeg for their critical yetconstructive comments. This work was financially supported by The Chinese University of Hong Kong ResearchCommittee Funding (Direct Grants) (Project Code: 2080082). The errors and inconsistencies remain our own.

Appendix. Open-ended questions of the questionnaire

(The original questions were in Chinese. The following are English translations of the questions.)

1. Why did you decide to become a mathematics teacher?2. What kind of qualities do you expect to nurture in your students?3. Some students marginally handle unfamiliar topics in mathematics curriculum such as

algebra but find that they do not really need such mathematical knowledge in theiroccupation and daily life. Furthermore, some students do not have interest and do notunderstand what they learn in mathematics but still can get a quite good academic result.How do you face these phenomena in your teaching?


4. Some people think that probability is not suitable to be taught to students: dices andplaying cards are usually used as examples and hence it is a gambling-related topic.Suppose the Education Bureau asks your opinions on adding or removing topics inmathematics curriculum. How will you react?

5. Are there any mathematics topics or mathematics teaching methods which are in conflictwith your religious beliefs? If so, how would you handle them?

6. Are there any mathematics topics or mathematics teaching methods which can demon-strate your religious beliefs? Can you give some examples as an illustration.

7. Do you think there is any difference between a teacher who believes in X and one whodoes not believe in X? (Remark: X is the religion which the respondent subscribes. For therespondent who does not subscribe to any religion, the question is read as: Do you thinkthere is any difference between a teacher who subscribes to a religion and one who doesnot?)

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Documents

Worldviews, religions, and beliefs about teaching and learning: perception of mathematics teachers with different religious backgrounds