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Montessori mathematics in early childhood education

Nicola Chisnall and Marguerite Maher
Abstract: 

The Montessori movement recently celebrated a century of international education, spanning from early childhood through to tertiary experience. The first Casa dei Bambini, or children’s house, was opened in Rome, Italy, on 6 January 1907, and within three years the influence of Montessori education began to reach New Zealand shores.

This article outlines the Montessori approach to early childhood curriculum in general, and discusses findings from a small research project examining mathematical concept development in children prior to school entry. Initial findings of the project indicate that the Montessori approach may have a positive impact on children’s numeracy knowledge and strategies at age five.

Montessori mathematics in
early childhood education

Nicola Chisnall and Marguerite Maher

Abstract

The Montessori movement recently celebrated a century of international education, spanning from early childhood through to tertiary experience. The first Casa dei Bambini, or children’s house, was opened in Rome, Italy, on 6 January 1907, and within three years the influence of Montessori education began to reach New Zealand shores.

This article outlines the Montessori approach to early childhood curriculum in general, and discusses findings from a small research project examining mathematical concept development in children prior to school entry. Initial findings of the project indicate that the Montessori approach may have a positive impact on children’s numeracy knowledge and strategies at age five.

This research arose from the involvement of the authors of this paper in the development and delivery of teacher education degrees at early childhood education (ECE) and primary levels that include a Montessori specialty in their final year. Our experience in sharing both the Numeracy Development Project and the Montessori mathematics curriculum has resulted in many discovery moments for our students. This has led us to suggest that wider understanding and dissemination of Montessori curriculum ideas may help to progress discussion on early mathematics development.

Introduction

This year (2007) marks the centennial of the founding of the Montessori education approach. Montessori is an education and peace movement, which reaches from early childhood to tertiary education in many countries and communities throughout the world. It has, nevertheless, been largely absent from curriculum discussion in New Zealand. In this paper we introduce readers who may have a passing knowledge of the Montessori system to some ideas that may contribute to curriculum debate in New Zealand, and particularly to the topic of early mathematics.

Shortly after the opening of the first Casa dei Bambini in Rome, word of Montessori’s children reached New Zealand shores. Schools were quick to see the possibilities of the Montessori child-centred approach, but a selective adaptation of the philosophy led to an incomplete replication of the original results, and in less than 10 years most had moved on to other ideas (Miltich-Conway & Openshaw, 1988; Shuker, 2005). Following a revival of Montessori education in both the United States and Europe, a second wave of interest began in New Zealand in the mid-1970s and there are now nearly 100 early childhood centres, a further 30 primary classes (often within state schools), and two secondary schools that draw on the Montessori approach.

What is the Montessori approach?

The Montessori early childhood centre is sometimes described by its original name, Casa dei Bambini, which means house of children. In this place, which ideally has a home-like atmosphere where all the fittings and fixtures, tools, and activities are adapted to the size and capabilities of the children, the goal of the teachers is simply to assist the child to reach his or her potential. They provide children with the means to help themselves by patiently modelling many self-care activities. In addition, as they work with a multi-age group spanning a three-year age range (ideally from three to six), a social community is created where mutual respect and collaboration are fostered. Older children are able to share new-found skills in helping the younger ones, and the younger children frequently observe and absorb new ideas from the activities of the older ones.

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Children in a Montessori early childhood centre have access to many precisely formed materials, enabling visual and hands-on exploration of concepts such as the relationship between size, quantity, and form. The founder of the approach, medical doctor and anthropologist Maria Montessori, believed that the child in the early period of life learns best when all the senses are stimulated. They begin, however, with practical care, related to dressing, eating, caring for others and their classroom or centre— a much trickier prospect in the days of buttoned boots, bows, and bonnets, but still one that offers challenges today as children master shoelaces, sunscreen, lunchboxes, yoghurt pottles, and other fancy packaging.

In a Montessori classroom, children then move on to sensory materials such as the cylinder blocks where, if one cylinder is placed in the wrong hole, the final one does not fit. In this instance the material gives the feedback and the child, remaining in control, will persevere to find a solution. Later a remembered pattern becomes the guide, and quite soon imagination leads the way to new solutions. As Montessori points out, people “do not produce with their feet and bodies, but with their spirit and intelligence, and when these shall have reached the level of development which is proper to them, then all our ‘insoluble problems’ will have become solved” (Montessori, 1949/1988, p. 85).

Further materials give children opportunities to experience ideas through sensory exploration, such as rough and smooth (tactile), sweet and sour (gustatory), loud and soft, high and low (auditory), warm and cold (thermal), heavy and light (baric), dark and light (chromatic), floral and spicy (olfactory), and many gradations in between. These sensorial tools were envisaged by Montessori to be a bridge to abstract understanding and reasoning, and are thus termed “materialised abstractions”.

Originally, the provision of practical and sensory education was as far as Dr Montessori intended to go with her group of 50 children. However, first the children and then a small delegation of parents approached her to see if she would show the children how to read and write. At first she refused, thinking that the children were still too young, but then she considered what would now be termed the “cultural capital” of the deprived children in her care, and this helped her to reach a decision to see what she could do while still continuing with her multisensory approach. She created a set of sandpaper letters that enabled the children to see, touch, and hear the alphabet sounds as the teacher modelled them for the children. The children quickly began to utilise this phonemic knowledge in an explosion of writing and, through this surprising activity, Montessori’s international reputation began to grow. Many visitors began to trek to Rome in a bid to find out more about her ideas.

In New Zealand, the importance of experience and exploration with curriculum materials in the early primary years has led to an explicit approach which aids the understanding of teachers and has led to much improved support of children. The Montessori approach has a similarly systematic approach, beginning in the early childhood years. Although Montessori’s so-called “scientific pedagogy” may be seen as a challenge to the strong emphasis on sociocultural learning in current New Zealand ECE practice, those who are familiar with the Montessori approach would argue that it originally drew knowledge and strength from the demands of the first parents and children in the slum district of San Lorenzo in 1907. This pedagogical approach went on to respond and develop, according to the social and cultural framework, wherever it found itself.

Montessori’s early emphasis on sensory- and material-based learning as a tool for early cognitive development is combined with an approach that acknowledges the importance of free activity and natural collaboration. This may be allied to the work of what Roth and Lee (2007) have termed “third generation” activity theorists such as Yrjo Engestrom, Jean Lave, Barbara Rogoff, and James Wertsch. Cultural-historical activity theory argues that “learning is equivalent to the mutual change of object and subject in the process of activity” (Roth & Lee, 2007, p. 198), a reciprocal process that leads to transformation in both social and material worlds. Today, Montessori programmes that are sensitive to their history continue to set store on contributions from the local family, community, and cultural context, but also see the child as a prime agent in cultural cohesion and adaptation (Montessori, 1949/1988).

In addition, the developmental perspective has not been lost in Montessori education. Baker (1991), for example, explains the process of “developing the mathematical mind”, pointing out that the difference between children in the first and second “plane” of development (the first plane is birth to around six and the second plane is from six or seven to around 12) is that whereas “in the first plane the child pays attention to the organization and classification of objects, the child in the second plane pays attention to the organization and classifications of abstractions” (p. 81). It is for this reason that a material-based curriculum is seen as so important in the early years.

Montessori developed a similar approach for numerical knowledge: wooden rods for counting from 1 to 10 were thus associated with sand-paper numerals, plus a further game to aid discovery of the concept of zero. Montessori then developed material to introduce the decimal system. This was originally reserved for primary-aged children but, once again, the younger children demonstrated their interest and power to absorb these concepts and so they too got to utilise the “golden beads” approach to mathematics.

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In the photograph above (Figure 2), bead materials are used to represent 1 (unit bead), 10 (bar), 100 (square), and 1,000 (cube). A size, weight, and form differential enhances the learning. Montessori employed her early love of mathematics to devise a further range of didactic materials to help children understand the operations of addition, multiplication, subtraction, and division, and later, for older, primary-aged children, fractions, cube root, algebraic formulae, and geometric relationships.

We have been intrigued to find that many of our teacher education students, in both primary and early childhood, find that the games played with these concrete materials provide eureka moments in discovering and understanding the concepts that underlie long division, algebra, and geometry. They frequently express the wish to have learnt with Montessori materials in childhood so that their uncertainty in mathematics could have been forestalled.

An example of the stepwise Montessori approach may be found in the following example of the trinomial cube (see Figure 3), which follows on from sensory activities including the experience of stacking, comparing, and classifying a variety of geometric solids.

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This wooden puzzle, representing the expanded formula of (a+b+c)3, is first explored visually (usually by children from around the age of four) using the coloured faces of each square as a guide to matching as they are reassembled (the largest “a” cube is red; b is blue; and the smallest, c, is yellow). Later, a plain, unpainted version is made available so that pattern, size, and shape become the predominant relationships. In the first primary class (for six- to nine-year-olds), children discover the theorem by deconstructing the cube and labelling each rectangular prism or cube block. Thus: (a+b+c) × (a+b+c) × (a+b+c) is revealed as a3 + 3a2 b + 3a2 c + 6abc + b3 + 3ab2 + 3b2c + c3 + 3ac2 + 3bc2. (Figure 3 shows the three stages of the algebraic process. The expanded formula may be deduced from the arrangement of the columns.)

Theoretical background

When we search for background on these and other materials, we find that the literature in the Montessori field has often been descriptive, with research primarily looking at the difference in curriculum opportunities provided for students.

Helfrich (2002) outlines the Montessori mathematics curriculum and suggests that one of the characteristics of the approach is that “throughout the children’s experiences in the Children’s House, they are assisted in the process of attaching precise and accurate language to both the objects they experience and to the activities themselves” (p. 11). This emphasis may provide one key to the Montessori experience, and Lillard (2005), citing research by De Loache, develops this theme when she notes that “when an object is both a symbol and something to play with, children have trouble seeing it as a symbol” (p. 64), and suggests that Montessori centres may well be on the right track in not mixing toys and teaching materials.

A series of North American studies has attempted to compare Montessori to traditional schooling, with mixed results. An early study by Bauch and Hsu (1988) explored the difference between Montessori and Piagetian theory with regard to the development of number concepts. They concluded that Montessori’s concrete materials, which often have a feedback mechanism as part of their design, have stood the test of time and enable children to surpass those in “traditional nursery schools on Piagetian seriation tasks” (p. 116). Reed (2000) studied place-value understanding by comparing Montessori and non-Montessori elementary classes (Grades 1 to 3) and found statistically significant differences favouring Montessori students on conceptual tasks.

Chattin McNichol (1992), however, cites Baker’s 1988 study of child interpretation of subtraction and points out “we don’t really know how deeply Montessori preschoolers understand the concepts embodied in the math work they do” (p. 114). Chattin McNichol notes other research (including the longitudinal Miller and Dyer study) that shows positive long-term effects on mathematics scores following exposure to a Montessori programme, but concludes that we “clearly, need more research on what Montessori children understand from what they do with the math materials” (ibid).

Not all studies have been positive. Cisneros (1994) found no significant differences between third-grade students in a matched study of public school Montessori and traditional schools. Fero (1997) found second-grade students “from traditional classes scoring significantly higher than students in Montessori classrooms in mathematics computation and mathematics concepts and applications” (p. x), although his comparison for all subjects rated Montessori student achievement at a higher level overall once students were matched for a range of variables such as aptitude, grade, gender, and socioeconomic status.

A more recent study carried out by Dohrmann (2003) compared students who had attended Montessori programmes from age three to 11 in the Milwaukee public school system with a control group (matched for gender, race/ethnicity, and socioeconomic status) of regular education students. The research was carried out using standardised tests five to seven years after the children had exited from the Montessori programme. Montessori students gained significantly higher mathematics and science scores than the regular education students, indicating that the mathematics programme in Montessori may have a long-term impact on student success.

The most recent study from the United States, Lillard and Else-Quest (2006), involved comparing a Montessori inner-city public school group with a matched control group, where all children were selected by lottery. These results also indicated a more positive outcome for the Montessori children in mathematics and reading, social skills, and executive function at the completion of their kindergarten (Year 1) year. Further anecdotal evidence has been reported from a trial in Gorton Mount state school, a school that was previously under “special measures” in a deprived area of Manchester, England. After one term of a jointly funded Montessori and government intervention at the foundation stage level, the children were described in a spot check by an Ofsted inspector as “at ease with their learning”, demonstrating “very high levels of perseverance and concentration”, and in a “calm working environment”. The mathematics scores of these children reached the level of the national average for the first time in the history of the school (Williams, 2006). The associated research report on this intervention is still to come.

In an effort to bring a local dimension to these comparative studies, the authors decided to carry out a small study focused on examining numeracy concept development in Montessori and other ECE centres. It was decided that the Numeracy Project Assessment (NumPA), developed as part of the Numeracy Project, would be a suitable standardised tool to assess the significance, if any, of the Montessori programme for the development of mathematics knowledge and strategies at the early childhood level. It should be noted that the Montessori early childhood programme ideally continues through to the end of the fifth year with a much greater focus on mathematics and language activities during that final year. Given, however, that most children enter school at age five, we decided to see what impact the Montessori emphasis on systematic concept development using sensory and manipulative exploration has on children up to the age of five.

Background to the Numeracy Project in regular education in New Zealand

In the late 1980s and 1990s the poor performance of New Zealand students in comparative international studies led to research on the best way to meet the mathematics needs of children in their early years of school, with particular focus on the lowest quartile. These studies showed that children who have limited basic mathematical concepts on entry to school are unlikely to make rapid improvements, as demonstrated by studies on stability of performance (Young-Loveridge, 1991). Nearly a decade earlier, in his longitudinal study, Newman (1984) found that students’ levels of achievement at age 15 could be predicted just as accurately from performance at age seven as from performance at age 10. We suggest that if there is stability of performance between the ages of seven and 15, then clearly what happens prior to seven is crucial.

Initially, in New Zealand, investigation was focused on the viability and success of early mathematics intervention (Young-Loveridge, 1993). At the same time, research in Australia studied both remedial programmes and then the more inclusive Count Me In Too (CMIT) programme, which was introduced in New South Wales government schools from 1996. Positive evaluations of the CMIT programme (Bobis & Gould, 1998) led to greatly expanded implementation, and in 2000 it was implemented in 81 New Zealand schools on a trial basis (Wright, 2002). During the late 1990s and early 2000s the Early Numeracy Project in New Zealand was developed, building on the CMIT programme.

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Fundamental to the Numeracy Project is the realisation that most children go through a predictable progression of stages in number concept development. These stages have identifiable, definable, and describable characteristics. This was a revelation to most teachers in the early years of schooling, and it has revolutionised the way mathematics is taught. The Early Numeracy Project describes the Number Framework or “Number Mountain”, which is depicted as an inverted triangle showing the increasingly sophisticated strategies employed as students progress and their knowledge increases (see Figure 4).

In the Numeracy Project there are two clear elements to children’s ability in number: knowledge and strategy. The Numeracy Project works from the premise that children use strategy, must be given ample opportunity to develop strategy, and having used the strategy enough the concept thus developed becomes knowledge. A further assumption is that teaching can be specifically targeted to encourage children’s advance through the strategy stages of the Number Framework. The question therefore arises as to whether children are experiencing both sufficient exploration and explicit direction in early childhood to enable the development of beginning strategies and knowledge in the mathematical sphere.

In contrast to the Number Framework, we suggest that children seldom start at point zero regarding numeracy concepts at age five, because a whole range of experiences and sensory perceptions form the foundation for numeracy concepts in ECE. An ever-broadening spiral process might provide a better representation, wherein knowledge feeds strategy trials, which lead to concept development, which creates new knowledge. Figure 5, however, acknowledges the ECE experiences built upon in primary schools through the Numeracy Project. As noted above, Montessori experience tends to challenge the Piagetian framework, and if we open up the lower point of the triangle, teachers may find that the possibilities for learning during the early childhood foundation period, prior to the “official” conceptual start, may be surprising.

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Current research

There has been limited international research and no New Zealand-based research on the effect of the Montessori approach in mathematics. Our research project was planned to inform the Montessori movement and those running teacher education programmes at Auckland University of Technology. The researchers hoped that if the findings of the pilot study were positive, it would provide useful evidence of an approach that could be shared with other education providers. If the results were negative or neutral, the researchers hoped to discover in what areas they could suggest improvement in the experiences offered in Montessori early childhood centres and schools.

Design

For this research project we adopted a quantitative research paradigm, utilising the Statistical Package for Social Sciences (SPSS) for analysing the data.

The participants were children within two months of turning five, or in their first month after turning five. The instrument used was the Numeracy Project Assessment (NumPA) (Ministry of Education, 2005). This is an assessment carried out with all children on entry to formal schooling at schools where the Numeracy Project has been implemented, and aims to discover at what strategy stage they are functioning and what level of knowledge is evident in the following domains: addition and subtraction; forward number word sequence; backward word number sequence; number identification; and place value understanding. The NumPA was completed by the researchers with individual students.

Sample

A time-restricted sample was drawn from:

a.&;&;five-year-olds who had completed 18-plus months in a Montessori setting in a variety of decile areas, entering regular schools

b.&;&;five-year-olds who had completed 18-plus months in a Montessori setting and who will complete the three- to six-year cycle in Montessori centres

c.&;&;a control group of five-year-olds entering regular schools from similar decile areas (to the sample children in a. and b.) who have attended alternative/other ECE centres (e.g., kindergarten or childcare) for 18-plus months.

Findings and discussion

The sample of 62 children who participated in the study included 34 attending Montessori preschools and 28 attending other (kindergarten and childcare) centres. When comparing these children’s performance on the NumPA, there was no statistically significant association between ECE type and performance on the strategy stages, in the forward number word sequence (FNWS), the numeral identification, or the basic facts tasks. In two tasks, however, there was a significant difference between the Montessori children’s performance compared with the other children. Table 1 shows that in the backward number word sequence (BNWS) task 75 percent of children at other centres were at stage 0 or 1, as against only 48 percent of the Montessori children (p < .01).

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More than half the Montessori children were at stage 2 or above, as against only 25 percent of the others. Since BNWS is the precursor to subtraction and is problematic for some children, this result indicates an area for further investigation. The NumPA was unable to tell us, for example, whether it is the golden bead material and other associated concrete Montessori materials that are providing opportunities for supporting understanding of BNWS, or if Montessori teachers and/or parents/whānau are providing other learning opportunities that enable the development of this strategy.

In the place value task there was a further statistically significant result (p < .004) for the Montessori children. Table 2 shows less than 40 percent of children at other centres being at stages 2 to 3 on place value tasks, whereas over 80 percent of Montessori children were at stages 2 to 3.

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There is a plethora of literature (Cobb, Yackel, & McClain, 2000; Munn, 2006; Wright, 2000; Young-Loveridge, 2001) indicating that an understanding of place value is vital to children’s progress in numeracy acquisition. As demonstrated in Figure 6, most four- (and five-) year-olds in a Montessori setting will experience, or at least observe, work with the golden bead material, which offers opportunities to work through Bruner’s enacted, iconic, and symbolic representational stages (Bruner, cited in Marsh, 2004) of conceptual development in place value.

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These statistics are of further possible consequence in view of the socioeconomic status (SES)1 of the centres involved. This happened by accident. In order to find sufficient participants of the correct age in Montessori units, centres all over New Zealand were used, resulting in a greater percentage of low SES centres being selected. For expediency, children of the right age in other centres were chosen close to the university and were mostly high SES. This threw up interesting and unexpected data.

Figure 7 shows that two-thirds of the Montessori preschools are low SES and one-third high SES. Of the other centres, only one-quarter are low SES, and three-quarters are high SES.

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It is possible that children may have been transported into these areas, and in one case this occurred; however, two of the Montessori centres were sited in rural areas in Auckland and Wellington, and a further two were in South Auckland, where parents were typically of lower educational achievement. We acknowledge that a more rigorous design would help to control for this variable.

The literature suggests that there is a correlation between high SES and children’s high achievement in mathematics in the early years of schooling (Wright. Stanger, Cowper, & Dyson, 1996; Young-Loveridge, 1991, 1993). One would therefore expect that the children in the Montessori preschools, a high proportion of which were low SES, would be likely to perform less successfully on the NumPA than their peers from high SES centres. In our sample this was not the case. Indeed the reverse is true, highlighting that it could be the way numeracy concepts are introduced to three- to five-year-olds in early childhood centres that is important.

Indications are that those centres in the control group that had a particular focus on providing a range of numeracy activities had a similar result to the Montessori centres. A larger sample and more detailed interviewing of staff and parents would tell us whether this result is likely to be replicated in other centres with a similar focus. It would also tell us if our results were significantly affected by variables such as the types of experiences made available in early childhood settings, the highest qualification of parents, and the activities and opportunities provided in the home.

The results of the research project serve to indicate, if nothing more, that further investigation in New Zealand ECE centres and schools using the Montessori approach would be useful. We are aware that “teachers’ use of new materials and a new vocabulary, does not necessarily guarantee effective teaching” (Anthony & Walshaw, 2006, p. 8), but research into enacted curriculum has suggested that when learning contexts are “jointly constructed by teachers, students and materials”, learning is enhanced (Ball & Cohen, 1996, p. 7). Furthermore, Spielman and Lloyd (2004) contend that when teachers and student teachers have the opportunity to learn mathematics with innovative curriculum materials, their understanding is more likely to endure. The communication to children of the excitement and enjoyment of mathematics, made possible through the Montessori materials, may well contribute to this effect.

Certainly, as more teachers educated in this approach come on stream, Montessori education is likely to become more accessible, but the question remains concerning Montessori curriculum materials: How useful would they be to children in New Zealand early childhood centres and primary schools? Would more explicit exploration and guidance be helpful?

Te Whāriki, the early childhood curriculum of New Zealand (Ministry of Education, 1996), has not emphasised specific curriculum content in a bid to prevent push-down curriculum from the primary sector, although this has been the subject of some challenge. The importance of emergent literacy and recognition of support for literacy through social practice is now widely recognised by the early childhood field (Barratt-Pugh & Rohl, 2000; Hamer & Adams, 2003; Makin & Whitehead, 2004), but as yet a similar interest in and awareness of how to support emerging mathematics concepts is not standard in all or even many early childhood centres. Cullen (2003) has been at the forefront of suggestions that Te Whāriki is well positioned to include “the weaving of subject content and skills through interest-based learning” and that this idea “needs to be more explicitly acknowledged” (p. 285). She points out that there could be a place for more specific programmes in literacy, mathematics, or technology. A recent New Zealand research initiative on enhancing mathematics teaching and learning in early childhood settings by Haynes, Cardno, and Craw (2007) may be a signal that the time is right for more attention to be given to this curriculum area.

Conclusion

We anticipate that further research will indicate the areas where more explicit connections could be highlighted for children. The researchers then plan to develop a professional development intervention for teachers to give specific information and teaching strategies on how numeracy concept development can be enhanced using concrete materials available in the Montessori setting.

Montessori, as a philosophical approach, has become well established in New Zealand and continues to grow. The research serves as a timely reminder of the importance of education in the first five or six years of life. The findings of the current study, while interesting, are preliminary, but they focus us on the need to identify and disseminate a range of teaching approaches that may promote further exploration in the field of enhancing the learning of our youngest children.

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Note

1&;&;&;In New Zealand the word decile is used to describe SES.

The Authors

Nicola Chisnall is senior lecturer and Montessori co-ordinator of the Bachelor of Education ([Specialty] Teaching) at Auckland University of Technology. Her research interests are in Montessori education, teacher transformation, and social justice issues as they pertain to young children.

Email: nicola.chisnall@aut.ac.nz

Marguerite Maher is senior lecturer and programme leader of the Bachelor of Education (Primary [Specialty] Teaching) at Auckland University of Technology. Her research interests are mathematics teaching and learning with young children and inclusive education.

Email: marguerite.maher@aut.ac.nz

Acknowledgements

The authors gratefully acknowledge the assistance of the children and teachers in the centres who took part in our research. We would also like to thank the Montessori Association of New Zealand for granting us permission to use their photographs of children in Montessori centres in New Zealand. Finally, we appreciate the helpful comments from the reviewers in editing and restructuring this paper.