The aims and stages of this project
The issue of home and school mathematics
The impact of Papua New Guinea education
The difficulties in developing measurement knowledge
Previous studies on measurement
Appendix - Measurement questionnaireOwens, K., & Kaleva, W. (2008, July). Indigenous Papua
New Guinea knowledges related to volume and mass. Paper presented
to Eleventh International Congress on Mathematics Education (ICME11),
Monterray,
Owens, K., & Kaleva,
W. (2008, July). Cases studies
of mathematical thinking about area in Papua New Guinea. Paper
presented to annual conference of the International Group for the Psychology
of Mathematics Education (PME), Morelia,
Owens, K., & Kaleva, W. (2007). Changing our perspective on measurement: A cultural case study. In J. Watson & K. Beswick (Eds.) Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (pp.563-573). Sydney: MERGA.
Owens, K. (2006). Rethinking cultural research. Proceedings
of Third International Conference on Ethnomathematics, Auckland,
This page was last modified on 28 June, 2008
This research will establish a
framework for understanding the mathematics of learners in transition from
home cultures to classroom mathematics. It is a case study in
1. identify the different types of cognitive and physical strategies used by a culturally diverse sample of Indigenous Papua New Guinean societies in various cultural activities that involve comparing and measuring length (including distance), area, mass, and volume and associated concepts such as ratio within cultural contexts;
2. collect data from as many different languages as possible to confirm the appropriateness of analysis of the systems and strategies and to establish a database of length, area, volume and mass measurements;
3. relate the findings to our current Western understanding of how children develop fundamental concepts of measurement;
4. undertake an evaluation of teaching using this new knowledge to improve students’ learning in elementary school.
As a human activity mathematics is embedded in a culture and a study of this ethnomathematics provides the backbone stability for mathematics (D’Ambrosio, 1990; D’Ambrosio & Gomes, 2006; Gerdes, 1996; Kaleva, 1993, 2001, 2003; Matang, 1997, 2001, 2003; Nunes, 1992). All cultures are involved in the mathematical activity of measuring (Bishop, 1988) but Bishop (1978) showed that tertiary students from Papua New Guinean societies thought differently about mathematics. He concluded “there is more than one way of viewing the world, the mathematician’s view is a particular one … shaped by a particular culture, it assumes many cultural ‘supports’, and increasing our own awareness of these cultural supports will improve the ways we introduce learners to the mathematician’s world” (Bishop, 1978, p.90). Cooke (1990) and the Garma mathematics project (Thornton & Watson) provide pertinent examples.
All children come to school from homes which may have significant cultural, linguistic and symbolic capital that is not tapped in formal educational settings (Gee, 1992). Recent studies in mathematics education have identified the transitions from home to school as significant (Civil & Andrade, 2006), but many of these studies have focussed on the conflict especially of the school system and the disadvantaged. Many young people in the world are experiencing a dissonance between the cultural tradition outside the formal educational institution (for example in their home or workplace) and the educational institutions (de Abreu, Bishop, & Presmeg, 2006). Studies of cultural conflict of immigrant students “suggest that an appreciation, by teachers, parents and learners alike, of the cultural capital that these students and their parents do possess, would ameliorate some of the symbolic violence (and) … would aid learners in the construction of self images that are strong enough to undergird their shifts in meaning as they build new cultural capital in transition experiences” (Presmeg, 2006a).
Disassociated knowledge can be rationalised “At home I add, at school I multiply,” said Bishop’s (1978) Papua New Guinean interviewee when confronted with conflicting choices of ways of finding area in these two sociocultural contexts. The different cultural approaches to mathematics may cause conflicts for students. For example, Bishop (1978) noted that in some villages the size of a garden will be measured by the number of paces across and down the garden. To Westerners, this is a semi-perimeter. However, at school the same person will calculate areas as a product of the length and breadth (if it is a rectangle). Bishop’s example suggests why children from different backgrounds may have different perspectives on quantifying area. His concerns highlight the importance of taking into account the effect of cultural values on measurement concepts.
However, this rationalisation does not generate a coexistence productive of a strong understanding of area. More productive transitions are the focus of this research. There is evidence that inclusive partitioning, in which opposites coexist, preserves heterogeneity and strengthens the parts of the classroom learning environment (Valsiner, 2000). Nevertheless, recognising and valuing the cultural mathematics must be part of using this knowledge in school mathematics (see Gorgorió, Planas and Vilella’s (2006) street seller who did not value this mathematics as a school student). The possibilities are widened by associations that belong to another language and culture but students who are struggling to gain understanding in the second language may overlook the potential provided by the wider associations gained from their first language and culture (Bishop, 2006; Yushau, & Bokhari, 2005). Complementarity underlying explicit cultural interaction may be dependent on adequate language skills and cultural immersion in the first language. “Ways of acting, interacting, talking, valuing, and thinking, with associated objects, settings, and events (impact on) … the mental networks” that constitute meaning but can only be determined by ethnographic study (Gee, 1992, p. 141) because of their implicit manifestation (Bishop, 2006).
Students need to know how their
own mathematics is structured and to compare their own mathematical system
with others to resolve such conflicts (Matang & Owens, 2004; Matang,
2005). Such studies are particularly important when the cultural approach
differs markedly from the one taught in schools. Yupiaq in Alaska have improved
their performance on standard mathematics test questions from their use of
cultural mathematics topics (Lipka & Adams, 2004). One cultural topic,
fish racks, involved the use of measurement. Schools for street children
in
Pacific nations have been throwing off their colonial education for decades as they struggle to provide an education for their communities that values their long-standing and Indigenous knowledge, languages and cultures while providing a gateway to education that supports them in a global world. Since 1986, the Government of Papua New Guinea has built on the concept of education that is Papua New Guinean in perspective (Matane ministerial report; Litteral, 2001). In 1991 the review of the education system included vernacular preparatory (elementary) schools in the formal system (pre-elementary, elementary grade 1 and grade 2) in order to improve an increase in access to initial education, with gradual introduction to English in grade 2 and a continuing recogntion of the cultural background of students through school. By 2000, this reform was being implemented with little understanding of the theoretical background for understanding their cultural mathematics and the links to western mathematics. Bourdieu (Fowler, 1997) has argued that cultural capital is a powerful tool for learning and social justice. Subtleties of language can impact on position and education. This is the case in PNG where respect impacts on the language of everyday activity and communication, knowledge is embodied in actions that are often observed and not described, and where certain people in a society have particular knowledge. At the same time, Lerman (2001) has argued that “children become mathematical by getting used to what counts as being mathematical, which is constituted in the social practices of the classroom” but if this is alien to the communication of the community then students fail to own and use their education as part of their citizenship. It is therefore a challenge for researchers to provide educators with tools by which they can consider how to make the tacit knowledge of the students explicit in their learning.
Students around the world find measurement concepts difficult (Zacharos, 2006). For example, area and perimeter are frequently confused by students (Hart, 1981; Doig, Cheeseman, Lindsey, 1995), repeated units and structure are poorly grasped (Outhred & McPhail, 2000; Outhred & Mitchelmore, 2000; Owens & Outhred, 1998, 2006; Willis, 2005) and proportional reasoning is often not achieved by students despite its fundamental position in mathematics (Behr, Harel, Post & Lesh, 1992). From international studies, only 41% of Australian eighth graders could compare time given in different units (TIMSS study); only 74% of fourth graders could say the mass of a brick stayed the same in different orientations (NCES, 2003) and only 29% could complete a diagram on grid paper to represent 13 square centimetres (ACER, 2002). The structure of measurement systems is frequently not recognised by students. Other writers have discussed the issue of estimation in measurement and shown its crucial role yet it is under-developed in most children and difficulty to teach (Boucher, 1998; Lang, 2001; Larkin, Perez, & Webb, 2003). Traditional communities are strong on estimation skills and this study should provide new insights.
This study of informal measurement systems will assist in recognising how intuitive understandings of measure can be transformed into structural understandings, as emphasised by Outhred and her colleagues (Curry, Mitchelmore, & Outhred, 2006; Kidman & Cooper, 1997; Outhred & Mitchelmore, 2000 & 2004). Key concepts of measurement are the units used, the ways repetition of these units are structured (Outhred & McPhail, 2000), and the need to compare quantities using identical units (a concept Piaget termed transitivity). An indication of increasing sophistication in thinking occurs when units are amalgamated to form composite units, and these new units are used. Willis (2005) points out that students and teachers may restrict their concepts and images of the abstract units for area by using concrete material tiles. Owens and Outhred (1998) illustrate the difficulties students have with representing tilings of areas but educators still need to understand the reasons for these difficulties.
Early studies on measurement related Piagetian conservation
tasks to performance on measurement tasks. While conservation frequently
accounted for a considerable amount of variance in scores on measurement
and transitivity, searches for other variables such as measures of information
processing capacity accounted for further variance. Nevertheless, these variables
and another variable logical reasoning with low correlation could not be
classified as prerequisites for learning about basic length measurement concepts
(Hiebert, 1984; Hiebert & Carpenter, 1980). Gal’perin and Georgiev (1960)
showed that students need to learn that a unit may have parts, a length may
be treated as a whole, orientation and visual comparison may need to be taken
account of in making comparisons, rearrangement may be used as a strategy,
identification of the attribute and of identity need to be established, and
comparison by use of units may need recognition of the the size of a unit.
Early studies in
This project builds on our earlier
projects. The Glen Lean Ethnomathematics Centre (GLEC), University of Goroka,
Measurement is more complex than counting systems. It is embedded in activities and is undertaken using tacit knowledge and visualisation. Explicating this aspect of measurement requires observation, discussion and reflection on practice. Counting may impact on the approaches taken. For example, groups of 20 paces or 4 banana bunches in a may be used as composite units if 20 or 4 are cycles in the counting systems. The cultural context will be influential and complex and yet it provides a richer basis for teaching. Every society compares and measures but how consistently, who measures, and more importantly why people measure must be understood for the transition from home to school to be made well. Accessing the measurement systems requires the skills of mathematics educators, first language speakers and an awareness of the possible impact of culture on the use of language in providing data. Tertiary first language speakers can assist with this research once they have recognised the invasion of their previous colonial education on their thinking (Bishop, 1988; Frade & Borges, 2005).
Use of informal units in village situations may influence PNG students’ understanding of measurement. A study of alternative cultural approaches will enable us to notice the cultural artefacts in representing measurement of length and related concepts such as area. For example, how does the village builder know how much is needed for a house when a floor plan is enlarged? This cultural lens, as Bishop (1978, 1988) declared, will assist us to understand students’ development better. Such a rich study of measurement of current cultures has never been attempted before and its insights will provide a theoretical perspective on how students learn to measure and understand measurement.
Our preliminary study (Owens & Kaleva, 2006) suggested that students were intuitively thinking of area in terms of length but not as a product as in school mathematics nor as a semi-perimeter (as Bishop, 1988 had thought) but as a way of counting fixed areas known from their context (e.g. standard garden widths). Recording, collating and analysing traditional strategies is essential before there is further loss of the many languages and cultures of PNG. The research will provide a sound basis for teaching in PNG elementary schools (beginning in their home language and transferring to English) so that PNG government funding for these schools is effective because the cultural context is addressed. Having a way of understanding their local mathematics is needed by teachers whose own schooling was often in English without a cultural focus. Mathematics educators including Kaleva and Matang (see papers on GLEC website, 2004) strongly support vernacular education and cultural mathematics as a good start to schooling. Older students need to know how their own mathematics is structured and to compare mathematical systems (NSW Board of Studies, Mathematics K-10, Stage 4; PNG Curriculum Reform, 2000).
This project requires a trans-disciplinary and trans-cultural group to carry out the research because the data collection, analyses, and theory require input from multiple cultures, mathematics educators and linguists. There are two approaches to this research. The first is to provide a record from as many language groups as possible. This data is often provided by more than one person. The method is firstly that of a survey and collation of materials. However, as analyses and issues arise the data collection has been modified. For example the questionnaire was modified to give greater importance to the linguistic aspects of the culture and further sampling has occurred in line with grounded-theory methodologies. This survey data is given meaning by the second more in-depth approach to the research involving an ethnomethodological approach. Observations and interviews are carried out preferably at the village. The choice of villages has been determined by participant researchers who are generally tertiary lecturers. The participant researchers have negotiated the visit and interviews generally with the assistance of others in the language group. Interviews with generally mature-aged tertiary students and staff has also provided rich data. Some interviewees were selected because of the richness of the information provided on the questionnaire or because they represented a remote group or one with a specific cultural aspect such as type of housing or terrain. An effort has been made to include both Austronesian and Non-Austronesian (Papuan) language groups and to select villages from different geographical areas such as mountain valleys, coastal mountains, bush and sago gathering communities, areas with kunai and those without, coastal villages that are remote from centres and villages that are closer to towns, and those in large river valleys.
A critical analysis has been made and during interviews and observations, notice was taken of contextual aspects that seemed to have a bearing on the discussions although they were not at first an ordered sequence of events or mathematically related. Where possible checks have been made of the data. If summaries were made from a number of questionnaires these were checked by someone from the language group. Village visit summaries were checked by the participant researchers.
Drawing this data together and preparing summary documents has been carried out using the provinces as the first breakdown (some language groups cross boundaries) and then by language group. Where there were comments from people who came from a different language group but were living or visiting there, their comments were noted separately.
Interwoven with this is an attempt to analyse measurement practices according to different variables such as terrain, linguistic and cultural features. There is some overlap in the sense that most Austronesian languages are found around the coastal regions where terrain will influence, for example, the type of house and canoe making.
Finally, this information will be used to provide example lesson plans for elementary and bridging classes and these will be used in a trial and evaluated before more widespread dissemination.
Stage 1
Collecting questionnaire data from tertiary students from a wide variety of languages provided a starting point for knowing the cultural activities that they see as relevant to measurement, how they perceive measurement, and what language words are available. This will be an on-going collection of data. The data from different people from the same language group will be compared. This data will be summarised under languages (using the SIL ethnologue) in provincial groupings. The questionnaire is attached. If you are a Papua New Guinean or know one of the languages we would appreciate if you could complete the questionnaire (electronically is good) and email it to us – email address on questionnaire.
In-depth interviews with staff, students and villagers in their villagers has provided rich information. Each interview was videotaped, audiotaped and/or dialogue recorded directly into the computer. These interviews in villages were associated where possible with observations of re-enactments of activities and visits to artefacts (eg gardens and houses). Villages have included a Western Highlands, Eastern Highlands, Markham Valley Morobe, and two Madang Province coastal villages close to Madang and one remote village and visitors to that village from the hinterland. Interviews have also been conducted with people from Simbu province (a male and a female), from Manus, from Western Highlands, and from a highland Madang village. In total 16 interviews have been conducted. In addition, we have held five focus group discussions (Smithson, 2000) taking students in one class and grouping them into two highlands groups, a northern coastal provincial group and a Papuan and islands group. This provided an opportunity to check information as well as to hear and compare practices from these groups.
There are extensive linguistic documents in PNG. These together with discussions with linguists have assisted in identifying and checking linguistic information. In particular, the adjectival words that are used with measurement, the word or sentence structure for comparisons, identified attributes, units and words for comparing and measuring in some languages have been identified whereas in other cases activities that might produce knowledge of attributes (area, length, volume and mass) and selecting of informal unit that can be used to establish the meaning of a unit and a composite unit for measuring that attribute have been identified. Most of this work is being gleaned from documents which were written for linguistic purposes so it is more by chance that information on measurement is gleaned from these sources.
The village data research will require entering the field with a younger member of the community who is familiar with western approaches to measurement but who can communicate with elders, observe activities and interpret the intentions of the actors. An ethnomethodological approach (Goode, 2003) will be used to give the community a voice. Part of the role of educating the PNG student researchers is to make them aware of the power relationships embedded in the research. That is the power relationships between themselves and the elders, that of the research situation itself, and the cultural expectations of research, researcher and the researched (Foucault, 1971; Holzknecht, 1987; Owens, 2006). Where possible videorecorded data will be used and later analysed with at least the student, myself and a PNG staff member of UOG and possibly students from the same or nearby language group. This will assist in interpreting the evidence. It will be a consensual discussion giving voice to the community and value to Indigenous knowledge. Field notes and audiorecording will be further used for triangulating the data.
All interviews are in Tok Pisin to communicate with village members and my 30 years of experience living and working in PNG cultures provided the expatriate researcher (Owens) with background and some knowledge of what I do not know as well as what to recognise as important and when to probe for further understanding. The research participants and negotiators frequently translate the questions into their own Tok Pisin or into Tok Ples. At times, lengthy discussions occur in Tok Ples with those sitting in the group before an answer is given. These are generally a result of the younger members of the community ensuring that they are sharing knowledge that is best understood by the elders of the community.
The anticipated methods for systematising and analysing these systems are (a) the theoretical approach formulated by Brousseau (1981) and (b) an open grounded-theory approach (Strauss & Corbin, 1990).
Brousseau (1984) identified three functions of knowledge corresponding to variations in the organisation of the milieu: implicit use (action situation); explicit use (formulation situations) and justification (validation situations). Applying this view to the PNG villager’s mathematical activities indicates that the context of the measurement application and the actions taken need to be analysed, as well as the justification of why this approach is used and what are the factors affecting the measurement.
The procedures and approaches suggested by Strauss and Corbin (1990) are used to develop the links between aspects of the measurement situation, systems and strategies. By using reflection, metaphor and similar procedures, it is anticipated that new insights will be gained into ways people are thinking in different cultures. The documents, focus group discussions, village interviews and observations will be the main data for analysis. As Strauss and Corbin explain it is important to vary the sampling throughout the project in order to assess and if necessary modify the emerging theoretical explanations. For this reason, villages and focus group discussions are being held over a period of years with on-going analyses.
The analysis covers (a) ways of comparing, (b) units, structures for repeating units, relationships between units, (c) tools, and (d) circumstances in which two attributes are measured simultaneously. Particular emphasis is placed on contexts in which length and area are used. These quantities are related in terms of the dimensionality of the units of measurement, so one focus of the analysis would be to examine the relationship between the dimensions of the unit of measure and the quantity to be measured. For example, are one-dimensional units commonly used to measure quantities in two and three dimensions?
Analysis considers links made between different measurement systems (e.g. metric, Imperial and cultural; school mathematics), counting systems (e.g. if composite units are groups of 20 when the counting system has cycles of 20) and generality of the attributes and units of length and area in different contexts (e.g for gardens and fish traps). A further strand of analysis is on the use of measurement language in metaphorical ways.
Stage 2
Tertiary students’ reports on ethnomathematics have identified ways in which cultural artefacts and practices can be associated with secondary mathematics. These will also assist to identify how measurement practices might be associated with classroom learning at elementary and primary school. In addition, activities using the counting systems of several language groups have been tried in elementary schools. These were based on activities from the Count Me In Project in NSW but with alignment to the PNG syllabus.
Building on these experiences, lesson plans will be prepared for schools focussing on number and measurement. These will be tried in schools in two ways. The plans will be adapted to the specific language group but they will also be used as guides for teachers in other language groups to how these teachers can adapt the ideas to their own language and cultural context.
By June 2008, we have collected questionnaire data from more than 200 people. We have interviewed 20 people from different provinces, language groups (both Papuan and Austronesian languages), and different terrains. We have collated document data, interview and questionnaire data on words related to measurement (see questionnaire for some examples). We have 315 records of words related to measurement. Some are from different people for the same language and they range from a couple of words from wordlists of large sectors of Morobe, Madang and East Sepik to records of 10 or more words and phrases.
Provincial summaries have been started with the descriptive questionnaire data forming the basis of the details for different language groups. In addition, summaries and additional notes are available for languages which were visited or for which interviews have been held.
Click here to transfer to the summaries page
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