ORIGINAL ARTICLE

Analysing the implemented curriculum of mathematicsin preschool education

Konstantinos Zacharos & Gerasimos Koustourakis &

Konstantina Papadimitriou

Received: 9 March 2012 /Revised: 1 March 2013 /Accepted: 14 November 2013# Mathematics Education Research Group of Australasia, Inc. 2013

Abstract The purpose of this paper is to contribute to development of research toolsfor observation and analysis of educational practices used by teachers in preschoolclassrooms. More specifically, we approached the implemented curriculum of mathe-matics in Greek preschool education. We analysed the recorded data from a week ofteaching practices in eight classrooms of Greek public kindergartens, based onBernstein’s theoretical framework on pedagogic discourse. The results showed thatthe actual educational practices in the observed classrooms deviated from the objectivesof the official new cross-thematic curriculum for teaching mathematics in Greekkindergarten in terms of the form of transmitted mathematical knowledge, the instruc-tional rules and strategies that teachers adopted for teaching mathematics, and theteaching–interactive relationships between preschool teachers and students.

Keywords Mathematics curricula . Preschool education . Pedagogical discourse .

Teaching practices

Introduction

Approaching and analysing the teaching of mathematics in a classroom is a subject ofeducation sciences, and, according to the research orientation, different aspects of teach-ing emerge. For example, sociological researches in South Africa showed that theimplementation of mathematics curriculum reforms is heavily influenced by social factors(Hoadley 2007; Vithal and Volmink 2005). Specifically, teachers in economically healthy

Math Ed Res JDOI 10.1007/s13394-013-0086-3

K. Zacharos (*) : G. Koustourakis : K. PapadimitriouDepartment of Educational Sciences and Early Childhood Education, University of Patras, 260.04, Rio,Patras, Greecee-mail: zacharos@upatras.gr

G. Koustourakise-mail: koustourakis@upatras.gr

K. Papadimitrioue-mail: kpapadimitriou@upatras.gr

areas are better educated and better informed of the objectives of educational reform. Onthe contrary, teachers in rural or poor urban areas have a lower level of education and havedifficulty understanding the teaching objectives of curricula (Vithal and Volmink 2005).Other studies place particular emphasis on highlighting the influence of psychologicalfactors in the implementation of educational reforms in mathematics. For example, theattitude of teachers towards mathematics (e.g., Peterson et al. 1989; Thompson 1992) orthe mathematics teaching efficacy beliefs (e.g., Philippou and Christou 2003) are param-eters that are considered to significantly affect the practices of teachers who are called toimplement the changes in mathematics curricula. Research on school teachers in Scotlandshowed that the implementation of mathematics curriculum has the special signature ofeach teacher (Macnab 2003). It is stressed in particular that “[i]t is their beliefs, practices,and working environment that shape and direct implementation” (Macnab 2003, p. 199).That is why most teachers (over 50 %) did not follow the changes suggested by thecurriculum concerning the teaching of specific topics of mathematics and the implemen-tation of a new teaching methodology (Macnab 2003).

There is little research regarding preschool knowledge and transmission practices(Skinner 2010; Tsatsaroni et al. 2003). More specifically, there has been no researchconcerning modern Greek preschool education that examines the relationship betweenmathematical curriculum and its implementation in preschool classrooms. A new post-PISA (Programme for International Student Assessment) 2000 curriculum reform wasestablished in 2003 for Greek preschool education (Alahiotis and Karatzia-Stavlioti2006), which introduces subjects such as Greek language, mathematics, environmentalstudies, creation and expression (i.e., arts, music, physical education), and informatics.This reform requires all of these subjects to be taught during the weekly teachingschedule (Ministry of National Education 2003).

The aim of this study is to investigate how teachers have implemented the newcurriculum for teaching mathematics in Greek kindergartens. More specifically, weattempt to explore the type of mathematical knowledge and the teaching methodschosen by the preschool teacher.

This paper begins with the theoretical framework and a description of the Greekkindergarten mathematics curriculum. Next, the research questions and methods usedare presented, followed by the research results and the discussion and conclusionsection.

Theoretical framework

We will rely on theoretical aspects of the work of Basil Bernstein and Paul Dowling toanalyse the educational practices in preschool classrooms. In particular, the theoreticalframework of Basil Bernstein (1990, 2000) on pedagogical discourse has proven to beuseful for the analysis of educational practices developed in the classroom for teachingmathematics (Adler 2001; Bartolini Bussi 2005; Fowler and Poetter 2004; Hoadley2007; Player-Coro 2011). This theory is useful as it provides descriptions and expla-nations of structuring the educational process and teaching of school knowledge, and ofanalysing complex forms of interaction in schools and classrooms (Apple 2002; Morais2002). The pedagogical discourse refers to “what” is transmitted (i.e., the content ofschool knowledge), “how” this knowledge is transmitted, and what skills it develops

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(Bernstein 1990; Morais 2002). Bernstein (1990, pp. 85–115; 2000, pp. 3–24) uses theconcepts of classification and framing to describe pedagogical discourse.

Classification of school knowledge

The concept of classification contributes to the analysis of the relationships betweencategories of agencies, agents, discourses and practices, because it “translates power atthe level of the individual (which) must deal with relationships between boundaries andthe category representations of these boundaries” (Bernstein 2000, p. 6). More specif-ically, classification in the case of school knowledge refers to the way that the contentsof curriculum relate to each other. Classification is considered strong (C+) where theboundaries between the different academic areas of the curriculum are clear anddistinct, whereas when the boundaries are blurred, the classification is consideredweak (C−) (Bernstein 1990, 2000). For instance, strong classification implies strictadherence to a specific scientific field. In this case, school knowledge is constructedthrough the recontextualisation1 of data from the specialised language and scientificpractices of the particular area, such as mathematics. In cases where classification isweak, interdisciplinary approaches are dominant and school knowledge is developedby the recontextualisation and combination of elements drawn from various academicareas of the curriculum.

Framing of pedagogic discourse

Framing refers to the forms of interaction that develop between teachers and students inthe classroom, and it describes the internal logic of the pedagogic practice. According toBernstein (2000, p. 12–13), framing refers to the degree of control that the teacher or thestudents have over the selection of the communication, the sequencing (what comes first,what comes second etc.), the pacing (rate of expected acquisition), the evaluation of theknowledge transmitted (criteria) and the control over the social base which makes thetransmission possible. Framing of selection and sequencing is strong (F+) when teachingis focused on the transmitter and students have limited choice in selection and sequencing.Framing of selection and sequencing is considered weak (F−) when teaching is focused onstudents and they have the choice of school knowledge to be transmitted and thesequencing, that is, the order of presenting and approaching the school knowledge(Morais 2002). A strong framing of pace (F+) means that the teacher determines therate of acquisition (Bernstein 2000). Framing of pace is considered weak (F−) whenstudents intervene in shaping the expected rate of transmission (Morais 2002). In this casethe preschool teacher gives some control to the students at the level of pacing. Criteriaverify whether the kindergarten students have acquired the knowledge that has been

1 The term recontextualisation refers to the selective transformation of knowledge from the field of itsproduction to the school context to make possible its teaching to students. It is about a process of selection,simplification, repositioning and refocusing (Bernstein 1990, pp. 191–192). More specifically, according toBernstein (2000, p. 113–114), the recontextualisation process “entailed principles of de-location, that is,selective appropriation of a discourse or part of a discourse from the field of production, and a principle ofre-location of that discourse as a discourse within the recontextualizing field. In this process of de- and re-location the original discourse underwent an ideological transformation according to the play of specializedinterests among the various positions in the recontextualizing field.”

Analysing the implemented curriculum of mathematics in preschool

transmitted. The framing of criteria is strong (F+), when the evaluation criteria are explicit,clearly defined and understood by the students. Framing of criteria is weak when theevaluation criteria are implicit to the pupils (Bernstein 2000). Finally, the control over thesocial base refers to the shaping of pedagogic interactive communicative relationshipsbetween the teacher and the students when teaching a subject such as mathematics.Specifically, the rules of social order refer to the forms that hierarchical relations takewithin the interactive framework of the classroom and to the expectations about conduct,character and manner (regulative discourse) (Bernstein 2000, p. 13). These relations areaffected by the instructional theories adopted for the teaching of a subject, which canmove between two extremes (Morais 2002). That is, the teaching can be entirely focusedon the transmitter, who, within the pedagogical interactive frame of the classroom, offersthe biggest possible control of the pedagogical act (strong framing). Also, the teaching canbe focused on the student and allow for a high degree of apparent control when reachingand conquering school knowledge (weak framing) (Morais and Neves 2011). Whenframing is strong, the rules of instructional and regulative discourse are explicit and“we shall have a visible pedagogic practice. Where framing is weak, we are likely to havean invisible pedagogic practice” (Bernstein 2000, p. 14). In this last case, the instructionaland regulative rules are implicit and, to a big extent, unknown to the students.

Distributing strategies and school mathematics knowledge

Dowling (1998) analyses the operation of the relay of school mathematics and ap-proaches the classification of knowledge putting emphasis on its forms which aredistributed through specific teaching strategies. He notes that “distributing strategiesmay expand or limit the range of message” (Dowling 1998, p. 115), because theyrepresent different forms of mathematical knowledge. Specifically, Dowling (1998, pp.145–149) notes that four distributing strategies can be applied for teaching mathemat-ics: generalising, fragmenting, specialising and localising.

The generalising strategies combine the use of “abstracting strategies” and “expandingstrategies” beyond the specific context of reference (Dowling 1998, p. 147). Also,Dowling (1998, p. 149) notes that when “the discourse is particularized, expanding […]strategies constitute fragmenting […].” Additionally, fragmenting strategies realise theesoteric domain,2 that is, the knowledge, principles, and practices of disciplinary knowl-edge, as segmental, rather than articulated. The generalising and fragmenting strategiesare within the scope of expanding strategies (Dowling 1998, p. 145).

A localising strategy exploits an empirical framework for teaching mathematics inorder to enable students to understand the specific school knowledge (Dowling 1998, p.285). Localising strategies refer to:

a) Non-mathematical knowledge, such as a story narrated by the teacher (e.g., a storyabout months) with non-mathematical content when teaching mathematics, and to

b) Actions of mathematical nature, which, according to Hoadley (2008), are distin-guished in the following categories:

2 According to Dowling (1998, p. 136), the esoteric domain “refers to the region of an activity which is mostclassified with respect to other activities. Both forms of expression and content are specialized.”

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– Nominal tasks, where students are asked to name various mathematical sym-bols or mathematical objects (e.g., to say the name of a number).

– Ritual tasks, where students are invited to participate in ritual activities, such asrepeating a word, phrase, or number all together after their teacher. In this way,preschoolers are not required to create something of their own.

– Mechanical tasks, such as copying numbers from the blackboard or colouringa drawing with mathematical content.

– Procedural tasks, where students carry out simple mathematical processes forthe approach and elaboration of mathematical knowledge (e.g., correspon-dences, writing mathematical symbols, equalities-inequalities, or writing math-ematical operations).

According to Dowling (1998, p. 147), specialising strategies can be used to letstudents enter the esoteric domain of mathematical knowledge. In the case of aspecialising strategy, principling tasks can be applied, which are complex teachingactivities that belong to the esoteric domain. The student must have special knowledgeof mathematics and have developed reasoning, justification, and explanation to ap-proach the principling tasks (Dowling 1998, p. 146–147).

Kindergarten mathematics curriculum in Greece

In 2003, the Greek compulsory education curricula were reformed. These curriculaincluded those for preschool, primary, and senior secondary education, and were called“cross-thematic” because of the objective to combine the teaching of every schoolsubject with knowledge from other subjects in the curriculum (Alahiotis and Karatzia-Stavlioti 2006). With this reform a change in the objectives of Greek kindergarten isobserved, as the previous preschool curriculum pursued mainly the cultivation of basicskills that would lead to the development of preschoolers’ personality (Ministry ofNational Education 1989). With this modern curriculum we shift from a curriculum thatdevelops basic skills to a curriculum of knowledge acquisition drawn from the knowl-edge areas of language, mathematics, studies of the environment, computer science,and creation and expression, and which seeks to support the transition from preschoolto elementary school (Koustourakis 2013, p. 74; Ministry of National Education 2003).Thereby, with the reform of the contemporary Greek preschool curriculum, the teachingof mathematics, which is considered a key subject of the curriculum, is emphasised andapproached “cross thematically” (Alahiotis and Karatzia-Stavlioti 2006; Koustourakis2007; Koustourakis and Zacharos 2011). Indeed, the thematic approach of preschoolmathematics knowledge is applied to countries such as China, England and USA (see:Hirsh and Reyes 2009; Kulm and Li 2009; Kwon 2003; Li and Shimizu 2009).

Greek kindergarten is a half-day programme for 5-year-old children. Its new curric-ulum suggests that the teacher should implement teaching practices that enhance thescientific thinking and the active participation of students in inquiry-based knowledge.These procedures refer to controlling and testing hypotheses, data processing, andformulating appropriate questions to enhance children’s interest in science. Moreover,the proposed teaching method emphasises the cross-thematic approach of knowledgedrawn from different academic areas (Ministry of National Education 2003).

Analysing the implemented curriculum of mathematics in preschool

According to the official curriculum, Greek kindergarten should aim for the system-atic involvement of children in organised mathematical activities. The teaching ofpreschool mathematics focuses on the following list of concepts:

– The identification, naming, and classification of elementary geometric shapes,– The presentation and creation of symmetrical shapes,– The comparison and measurement of geometric figures,– The familiarisation with mathematical relationships (e.g., associations, classifica-

tions, and permutations),– The concept of the number and numerical symbolism,– The ordering of numbers from 1 to 10,– The realisation of simple exercises of addition, subtraction, multiplication, and

division with concrete objects,– The estimation of results, and– The development of problem-solving processes.

An appropriate learning environment should be set up that will encourage studentsto participate in the educational process, to continuously evolve their thinking, and togradually achieve the learning goals set by the preschool teacher (Dafermou et al.2006). Moreover, it has been proposed that the teaching of mathematical conceptsshould be based on childhood experiences. In this way, the simple mathematicalknowledge that students have could be expanded, and the ability to apply newmathematical knowledge acquired in new situations could also be cultivated. In fact,it is proposed that a learning environment for mathematical concepts should bedeveloped that stimulates children’s interest and encourages them to recognise andprocess the mathematical relationships encountered in preschool activities.

Inquiry questions—methodology

In this study, we examine the following research questions:

– What is the form of the mathematical knowledge transmitted in Greek kindergar-ten? In others words, are mathematical activities a separate teaching subject or arethey integrated into a cross-thematic approach?

– What are the instructional rules and the strategies that teachers adopted for teachingmathematics?

– What are the didactic / interactive relationships between preschool teacher andpupils in the teaching of mathematics?

The sample

The research sample consisted of 137 students. A study was conducted in eight publickindergarten classes in the city of Patras in Greece, where 137, approximately 5-year-old, students were enrolled. Thirty eight of them were of an average age of 5.3 yearsold, and ninety nine were of an average age of 4.5 years old. It should be noted that inGreece preschool education includes 2 years of schooling (K1, the younger children,

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and K2, the older children). In most preschool classes, K1 and K2 children are groupedtogether in one class. Most of the students were Greek; the few children whose parentswere foreigners were born in Greece and were fluent in Greek. The students’ familiesbelonged in working class and lower middle class, in terms of fathers’ and mothers’academic qualifications and occupations. The preschool teachers who taught thesekindergarten classes had 4-year university degrees in educational science as well as10–15 years of teaching experience. These teachers had also attended a short trainingprogram on the content and objectives of the new preschool education curriculum,which was put into practice during the 2006–2007 academic year. This training wasorganised by school counsellors and lasted 2 days. Also daily training sessions areconducted by the school counsellors at the beginning and the end of each school year,where the cross-thematic approach of kindergarten school knowledge is discussed.

Procedure for collecting and processing the empirical data

The aim of this research is to explore how mathematics is taught in Greek kindergartens.This is because distinct knowledge areas, such as language, mathematics, environmentalstudy etcetera, are for the first time introduced in the new kindergarten curriculum andcorrespond to the subjects of Greek primary school curriculum (Ministry of NationalEducation 2003). The method of non-participant observation (Cohen et al. 2004, p. 186)is used to collect the data. Specifically, the researchers observed the eight classrooms foran entire teaching week. This extended unit of time was chosen to allow for a full cycle ofthe weekly curriculum to be completed. We ensured that this observation did not takeplace close to national holidays or festivities, when the curriculum and interest often shiftsto highlight the themes and preparation of the holiday. It should be noted that noinstructions were given to preschool teachers and children, because our intention was tostrictly observe the everyday pedagogical practices in schools. In order to avoid a potentialbias from teachers (i.e., modification of their behaviour and mathematical teachingmethods), our observation consisted of what was taught during the school week. Theresearchers remained in the classroom and observed the entire educational process frombeginning to the end, taping the implemented teaching practices, and keeping detailednotes. Detailed transcripts were produced upon the completion of this study.

The preschool teachers were aware of the purpose of this study and after each dailyprogram they had discussions with the researchers to provide clarifications on thetopics they chose to teach and the teaching methods they practised. Also, a day beforethe recording of teaching begun, researchers sat at the back of the classroom andwatched the daily program. This was done for the researchers to get a first idea of theconditions prevailing in the classroom and for the teacher and students to familiarisethemselves with the presence of the researchers.

For the processing of the empirical data, we created an external language of description(Hoadley 2007; Koustourakis, and Zacharos 2011;Morais 2002). This consists of a patternof encoding that concerns the interaction between the theoretical concepts from Bernstein(1990, 2000), and Dowling (1998) and the empirical data from research. The researchmaterial was analysed using the quantitative and qualitative content analysis. In particular,texts illustrating the daily teaching activities were analysed in task units (Hoadley 2007,2008). Here, a task unit is defined as a complete teaching activity with a specific teaching

Analysing the implemented curriculum of mathematics in preschool

objective, which is associated with a very specific subject; for instance, an example of amathematical task unit would be the size comparison between objects (larger-smaller). Wethen selected the task units that presented teaching activities with mathematical content.Here we must point out that some of these task units are cross thematic, that is, themathematical knowledge is combined with knowledge from other knowledge areas ofpreschool curriculum, and other task units have strictly mathematical content.

The various task units were classified into categories of analysis which were chosento be independent, complementary, and supportive of each other. Collectively, thecategories help to compose an image of the pedagogical practices implemented inschools for teaching mathematics. These specific categories of analysis emerged takinginto account the empirical research material and the classification proposed byBernstein (1990, 2000), and refers to the theoretical part as well as the more specifictypes of school mathematics analysis proposed by Dowling (1998).

Classification of school mathematics knowledge

The following two categories of analysis emerged from the study of the research material:

C+: In this case mathematics is taught as a separate subject.C−: Cross-thematic activities for teaching mathematics. That is, mathematicalcontent knowledge is connected with the knowledge of other preschool curriculumsubjects to help students understand mathematical concepts.

Strategies for teaching school mathematics knowledge in preschool classrooms

The following strategies for the analysis of preschool mathematics tasks emerged fromthe study of the research material (Dowling 1998; Hoadley 2007, pp. 685–686).

Localising strategies: Nominal strategies: naming of mathematical symbols andmathematical objects.

Ritual strategies: repetition and imitation of the mathematical actions as given bythe teacher.Mechanical strategies: teaching activities of mechanical character, such as copyingnumbers from the board or painting a drawing with mathematical content.Proceduralising strategies: simple mathematical activities. These activities areaccompanied by clear instructions.

Specialising strategies: Principling strategies: knowledge of mathematical rules isrequired along with reasoning, justification, or explanation ability, as well as novelapplications of knowledge.

Instructional rules for teaching mathematics in preschool classrooms

The following instances of framing emerged from the research material:

Selection: F+: the preschool teacher selects the mathematical knowledge to betransmitted.

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Sequencing: F+: the preschool teacher determines the order in which knowledge ispresented (what comes first, what comes second).

Pacing: F+: the preschool teacher determines the rate of acquisition without takingthe pupils into account.F−: pupils have more apparent control over the rate of expected

acquisition.Evaluative rules: F+: criteria for assessing mathematical knowledge are clearly

defined.F−: criteria for assessing mathematical knowledge are not

clearly defined.

Didactic / interactive preschool teacher-pupil relationships

The following two cases of framing (Koustourakis, and Zacharos 2011; Neves andMorais 2001, pp. 232–233) emerged from the study of the research material:

F+: when the preschool teacher has the leading role in the teaching process and theintervention and participation of pupils is required in the act of teaching.F−: when emphasis is placed on the highest possible degree of students’participation in the educational process.

In order to ensure reliability, the task units were classified by the researchers intoeach of the above categories of analysis in two different periods of time, 1 month apart.A task unit was included in a certain analysis category if there were at least five pointsof agreement (the acceptable percentage of agreement being over 75 %) (Koustourakis,and Zacharos 2011; Vamvoukas 2002).

In the next section, we will present our research findings and our analysis. Thisincludes the aggregate results of the teaching activities identified in the eight kinder-garten classes in order to identify patterns in the teaching of preschool mathematics.

Findings

Table 1 shows the breakdown of the mathematical task units according to discipline. Asseen from Table 1, the teachers mainly taught mathematics through cross-thematicactivities (56 task units, 58.3 %). However, the percentage of task units through whichthe teachers taught mathematics as an autonomous subject is also significant (40 taskunits, 41.7 %). In the cases where teachers tried to primarily or exclusively teachmathematics, the construction of mathematical knowledge appears to be an importantteaching parameter.More specifically, in 58 task units (first three rows in Table 1, 60.6%),preschool teachers taught mathematics either independently as if it was an independentand autonomous subject of the curriculum (strong classification: 40 task units, 41.7 %) orthrough cross-thematic activities with language (14 task units, 14.6 %) and creativity andexpression (4 task units, 4.3 %). The number of modules which provided some form ofmathematical knowledge through the study of other subjects was also significant (38 taskunits, 39.4 %); in particular, in 28 language task units (29.8 %) and 10 environmentalstudy task units (9.6 %), teachers provided students with mathematics-related knowledge.

Analysing the implemented curriculum of mathematics in preschool

Consequently, a large number of the teaching activities had purely mathematicalcontent (40 tasks, 41.7 %), indicating a strong classification (C+). However, in a largernumber of teaching activities (56 tasks, 58.3 %), the teaching of mathematics wasmixed with other disciplines, indicating a weak classification for the pedagogicaldiscourse (C−). The first lesson extract (Extract 1), as shown below, belongs in thelatter case (interdisciplinary). In Extract 1, it appears that mathematical knowledge isassociated with the acknowledgement of time, that is social character knowledge (i.e.,dialogue 1.1) as well as with elements in language (dialogue 1.16).

Extract 1: Discussion on date (C−)

In this thematic unit, the discussion of the teacher with the children revolves around thedate.With reference to the date, the teacher seeks to introduce children to writing numbers.

1.1 T (Teacher): Do you know which year we are in now?

1.2 Students: Two thousand and ten.

1.3 T: Well done! How do we write two thousand and ten?

1.4 S (Student): Zero, one, and two.

1.5 T: Very good, but they are not in the right order.

1.6 S: Zero, and then another zero, and ten.

1.7 T: One and a zero.

1.8 S: Yes!

1.9 T: I will write one.

1.10 S: Zero.

1.11 T: Good.

1.12 T: Let’s see it all together now.

1.13 T: (Writes the date) All this, is read, two thousand ten.

1.14 T: Who wants to read this for me?

1.15 S: Me.

…

Table 1 Total task units withmathematical content

Disciplines Units withmathematicalcontent (%)

Mathematics (C+) 40 (41.7)

Mathematics (Language) (C−) 14 (14.6)

Mathematics (Creativity and expression) (C−) 4 (4.3)

Language (Mathematics) (C−) 28 (29.8)

Environmental study (Mathematics) (C−) 10 (9.6)

Total activities with mathematical content 96 (100)

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1.16 T: But in which month are we now? We said this on Friday. Wewrote that over there on the blackboard. (The word “January” is writtenon the blackboard.)

1.17 S: Two thousand eight.

1.18 S: Two thousand and eight has passed.

1.19 S: The two thousand twenty.

1.20 T: Two thousand twenty has not yet arrived. We said this is the year (shows“2010”). I’m asking you for the month.

1.21 S: Two thousand one.

1.22 T: Let’s see, what did I write here? (The teacher shows on the blackboard theword “January.”)

Another example of weak classification can be seen in Extract 2. In this extract,children are invited to count the animals displayed in a photograph during a lessonrelated to environmental studies.

Extract 2 Count animals (C−)

The teacher addresses a student:

2.1 T: How many animals are there?

2.2 S: Six.

2.3. T: Do you all agree?

2.4. Students: No.

2.5. T: How many are they?

2.6. S: Five.

3.7 T: Well, are there five or six?

2.8 T: (Teacher leaves time for children to think.) Well, come on now, let’s counttogether.

Moreover, an example of a strong classification of school knowledge ispresented in Extract 3, where the teacher introduces counting quantities andaddition to the children.

Extract 3: Count objects (C+)

3.1. T: How many are the boys in our team?

2.2. Students: Eight.

3.3. Τ: And how many are the girls?

3.4. Students: Ten.

3.5. Τ: And how many are you all, boys and girls?

Analysing the implemented curriculum of mathematics in preschool

Children counted themselves, and then were asked to find from a set ofnumerical cards the one showing their correct, identifying number.

Table 2 shows the distribution of teaching strategies selected by the teachers to relaythe mathematical knowledge.

As seen from Table 2, the majority of mathematics teaching activities fallinto the category of localising strategies (97 % of all task units). This meansthat the main goal of the teachers is to enable students to respond to thespecific tasks without having to possess special knowledge of mathematics; inaddition, these activities have a very specific application framework. Thelocalising strategies can be broken down into four categories, of which nominaland ritual strategies constitute the majority. This means that the mathematicaleducation observed in this study was limited to the naming of things ornumbers as well as the reproduction and repetition of words and proceduresas shown by the teacher. An indicative example of such a strategy is seen inExtract 4, where the teacher goes through a calendar and invites a student toread the dates (localising-nominal strategy).

Extract 4: Localising strategy example

4.1 T: On Friday, what was the date?

4.2 S: Fifth.

4.3 T: What was the date on Saturday?

4.4 S: Sixth.

4.5 T: Saturday was sixth. On Sunday, what was the date?

4.6 S: Seventh.

Specialising strategies were identified in 3 % of the cases in this study, inwhich the possession of special mathematical knowledge was required of thestudents in order to respond to teaching activities shaped by the teacher. Extract5 is an example of an observed specialising strategy. Here, the goal is to teachchildren to solve simple subtraction problems by counting their fingers(specialising-principling strategy).

Table 2 Strategies adopted for teaching preschool mathematics

Instructional strategies Task Units

Localising strategies Mathematical Nominal 37 (39 %)

Ritual 29 (30 %)

Mechanical 6 (6 %)

Procedural 21 (22 %)

Specialising strategies Principling tasks 3 (3 %)

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Extract 5: Specialising strategy example

The preschool teacher poses a subtraction problem and invites children to use theirfingers to determine the result.

5.1 T: You watch me now. If from this bowl (showing a container with markers) Iremove four, how many markers will be left in the bowl?

5.2 Some students: We don’t know.

5.3 T: Show me with your fingers how many markers are in the bowl? I said therewere ten.

5.4 Some students: This many markers. (Students show fingers of both hands.)

5.5 T: If we close four fingers, how many will stay raised?

(Many children closed four fingers.)

5.6 T: How many fingers are raised?

5.7 S: (loudly counting their fingers) One, two…six. Six markers!

Table 3 shows the distribution of the task units in the category of framing.As seen in Table 3, teachers choose to teach mathematics at a strong framing of

selection and sequencing and a strong framing of pace. This means that the preschoolteacher decides solely which mathematical knowledge will be transmitted in his classand the order by which it will be presented to the students (96 task units, 100 %). Thisis clear from all the extracts in this paper. Also, the teacher monitors and identifies withhis instructional activities the time devoted to approaching the school mathematicalknowledge (F+ of pacing: 82 task units, 85.4 %). A typical example of strong framingof selection, sequencing and pace is shown in Extract 1. The content of this extractclearly shows that the teacher chooses the topic to be transmitted and the order ofpresenting the knowledge to the students, and identifies and monitors the time spent bystudents to approach the knowledge. The same applies in Extracts 2 and 3.

In terms of evaluative rules, in almost all teaching activities, strong framing (96.9 %)is implemented; teachers determine with great clarity the path students should follow inorder to arrive at the correct answer. For example, in Extracts 1, 2 and 3, the teacher,through continuous assistance and guidance, seeks to lead students to the right answer.

Table 3 Framing of task unitswith mathematical content

Framing Task Units

Selection F+ 96 (100 %)

Sequencing F+ 96 (100 %)

Pacing F+ 82 (85.4 %)

F− 14 (14.6 %)

Evaluative rules F+ 93 (96.9 %)

F− 3 (3.1 %)

Didactic / interactive relationships F+ 92 (95.8 %)

F− 4 (4.2 %)

Analysing the implemented curriculum of mathematics in preschool

Finally, in almost all of the task units, as shown in Table 3, a strong framing ofdidactic / interactive relationship exists between the teacher and the pupils (92 taskunits, 95.8 %). This can be explained by the way that the teachers ask continuousquestions and make many interjections, leaving no room for students to act indepen-dently and to develop autonomous actions to obtain and understand the mathematicalknowledge. We found that teachers seem to follow a more teacher-centred way ofteaching by choosing the kind of knowledge that is to be transmitted, as well asdefining the selection, order and pace of its presentation during the teaching process.

Discussion and Conclusion

The purpose of this research was to analyse the teaching methods of the appliedmathematics curriculum in Greek preschool classes. The modern Greek kindergartencurriculum specifies that mathematical knowledge must be transmitted through cross-thematic teaching activities. Also, the teaching practices recommended by the officialcurriculum emphasise constructivist teaching methods, enhancing the students’initiative and active participation in the construction of new knowledge, whilethe application of invisible pedagogies are proposed, where students are at thecentre of the educational process (Dafermou et al. 2006; Ministry of NationalEducation 2003).

According to Bernstein (2000), schooling institutions, such as kindergarten schools,constitute the field of reproduction of pedagogic discourse. In fact, the official peda-gogic texts in the recontextualising field of the classroom, the kindergarten mathematicscurriculum in our case, are transformed by the choices teachers make for shapingschool knowledge and by the teaching methodologies they use (Singh 2002). This iswhy the implemented curriculum is often differentiated from the intended, namely theofficial, curriculum to be implemented, as seen in our research. More specifically, theanalysis and elaboration of our research material that was related to the teaching ofmathematical concepts revealed several important points.

First, in Greek kindergartens, it appears that mixed options are used for therecontextualisation and shaping of mathematical knowledge. In particular, the classifi-cation of the pedagogical discourse reveals that the task units with purely mathematicalcontent represent a large proportion of the transmitted mathematical knowledge. Thismeans that despite the clearly stated objective of the new curriculum for the teaching ofknowledge areas / subjects through cross-thematic activities, this is only partiallyimplemented in the curriculum of the observed kindergarten classes.

The teaching of preschool mathematical knowledge used strategies which, accordingto the theory of Dowling (1998), correspond to particular forms of knowledge andreproduce features of specific domains of mathematical knowledge. In the majority ofthe instructional strategies, teachers selected localising strategies (97 %), in the form ofmainly nominal and ritual tasks. This implies that the transmitted mathematical knowl-edge is largely confined to naming practices (e.g., the pronunciation of the name of anumber) and to practical imitation of the teacher’s actions, which in fact does not helpstudents to inquire deeper into mathematical concepts. Also, the specialising strategiesused in the study were characterised by principling tasks (3 %), which indicate that thestudents participate in solving simple problems.

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Then, contrary to the objectives of the official kindergarten curriculum, the results ofthis research showed the apparent control performed by the teacher in all aspects offraming. In this case, the preschool teachers design and carry out the teaching ofmathematics, preventing the participation of students in what will be transmitted (F+

of selection 100 %) and the order in which order it will be taught (F+ of sequencing100 %), and determining the time that will be devoted to teaching a particularmathematical concept (F+ of pace 85.4 %). Also, the results show that the framing ofevaluative rules is strong (F+ 96.9 %), a fact which indicates that teachers clearlyidentify the mathematical issue at hand in each task unit.

The discussions that were held between the researchers and the preschool teachersafter the end of each daily teaching program showed that the teachers’ choices forteaching mathematics were influenced mainly by the following factors: first, thepressure of the educational culture upon teachers, which seems to be largely acceptedby teachers and parents, to prepare students for the next school level (elementaryschool), giving them basic knowledge of language and mathematics. Furthermore,teachers’ choices were influenced by the highly respected status of mathematics inthe new kindergarten curriculum. Another factor that appears to influence the teachingpractices of teachers and which needs further investigation is the socioeconomic statusof students’ families (Hoadley 2007, 2008; Koustourakis et al. 2014). Thus, particularpreschool teachers believe that the teaching of preschool mathematics with a directionaland steering way associated with the adoption of performance pedagogic models(Bernstein 2000, pp. 44–50) meets better their expectations to help their studentsunderstand the mathematics knowledge that has to be taught according to the newkindergarten curriculum.

Also, the choice of visible pedagogies for teaching mathematics derives from thequality of didactic / interactive relationships between teachers and students, wherestudents’ control is explicit. More specifically, the majority of the task units for thespecific case have strong framing (F+ 95.8 %). This means that the absolute control ofthe shape and direction of the educational process resides with the teachers, leaving noroom for students to act independently in terms of obtaining mathematical knowledge.This finding means that the implementation of the mathematics curriculum in thestudied schools deviates from the guidelines given by the new curriculum, whichstates that teachers should support initiative taking and the growing autonomy ofstudents in the learning process.

Although the results of this study can not be generalised, they are an indication thatthe educational method for mathematics, one of the basic subjects of the Greekcurricula, is treated by preschool teachers as having clear scientific frontiers (C+). Forthis reason, the observed teachers attempted to teach mathematics in the traditionalperformance model (Bernstein 2000; Morais and Neves 2011), where the emphasis ison the production of appropriate mathematical texts, by implementing choices on anexplicit and visible pedagogy (strong framing). This is a clear differentiation from theinquiry-based education practices as promoted by the new preschool curriculum forteaching mathematics (Dafermou et al. 2006). However, dealing with the differencesbetween the intended and implemented curriculum revealed in our research can be donethrough a conscious and systematic personal effort of preschool teachers, in order toharmonise the strong classification of academic mathematic knowledge with the weakclassification and framing common in preschool caring, as shown by the findings in

Analysing the implemented curriculum of mathematics in preschool

Tsatsaroni et al. (2003). Finally, we note that an interesting extension of thisresearch might be to investigate whether the implemented pedagogical practicesfor teaching preschool mathematics favour or not the students according to theirsocial and cultural background. This is an issue of concern to researchers whohave focused their interest on students who live in social environments with astrong tradition of racial and social differences (e.g. Hoadley 2007; Smith andSadovnik 2010; Vithal and Volmink 2005).

Acknowledgments The authors would like to thank the three anonymous reviewers for their helpful andconstructive comments.

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