Learning through play – pedagogy and learning outcomes in early childhood mathematics

ABSTRACT

Whilst research underlines the importance of early mathematics in kindergarten, practitioners need effective and innovative approaches to pedagogy. Currently, very different approaches are deployed from an instructional, educator-led approach based on training programmes to a play-based approach. This intervention study examines the effects on the mathematical competency of these two pedagogies. Thirty-five kindergarten educators and 324 six-year-old children were randomly assigned to either a training programme, a play-based approach with card and board games or to the control group. Educators’ views on the interventions were gathered in semi-structured interviews. The results indicate higher learning gains overall for the play-based approach. Differentiated effects were found as tendencies: children with low competencies tend to gain more from training programmes compared to no intervention; children with high competencies gain more from the play-based approach than the training. Educators evaluated the play-based intervention with card and board games as better suited to children’s diverse needs.

KEYWORDS:

• Early childhood education
• mathematics
• play
• quantity-number-competencies
• kindergarten
• games
• pedagogy

Introduction

Early mathematical competencies are highly relevant for later education outcomes (Duncan et al. 2007; Grüssing and Peter-Koop 2008). Whilst there is a growing awareness that children need to already be supported in their mathematical learning in kindergarten, there is little consensus about the best pedagogical approach. Kindergarten educators may emphasise that mathematical activities need to be embedded in everyday situations (Gross and Rossbach 2011) or that early learning needs to be based on play, even though the understanding of play itself varies (Gasteiger 2015). Furthermore, kindergarten educators might use a training programme for mathematics, to ensure that mathematical competencies are explicitly fostered. Little research exists on the effectiveness of these approaches, as they have not yet been systematically compared regarding the learning gains for all children and for outcomes of children with differing levels of competencies. The research project presented here compares the effectiveness of a play-based approach with card and board games (Hauser et al. 2015) with a training programme (Krajewski, Nieding, and Schneider 2007) with a control group. In addition, educators’ acceptance of an approach is important for effective implementation. Therefore, educators’ views need also to be taken into account. This paper addresses the following research questions: how does the play-based approach with card and board games compare with the training programme regarding children’s mathematical learning gains? Are there differentiated effects for children with differing mathematical competencies? What are educators’ experiences with and views on the play-based approach and the training programme?

Theoretical framework: mathematical competencies and approaches for kindergarten

The literature review starts with research findings on the relevance of mathematics education in kindergarten, followed by a focus on certain aspects of mathematical competencies. Thereafter, consideration is given to research on early childhood educators’ attitudes towards mathematics. Then, several approaches to early mathematics are outlined, followed by the discussion of the innovative potential of play in the teaching of early mathematics.

Mathematical competencies matter

Mathematical competencies at kindergarten are highly relevant for learning outcomes at school. In a meta-analytic regression, maths competencies in kindergarten, i.e. number recognition, number sequence, counting, ordinality, relative size, addition and subtraction were found to be the strongest predictors for later school achievement, reaching an average effect size of .34 compared with early reading (.17), attention skills (.10) and socio-emotional behaviours (no effect) (Duncan et al. 2007). Children with low mathematical competencies in kindergarten are most likely to experience difficulties with maths at school (Dornheim 2008). Quantity–number–competencies predict later mathematical competencies beyond number words whilst phonological awareness does not (Krajewski and Schneider 2009). Children’s mathematical competences differ considerably in kindergarten, which is also due to differences in the home learning environment (Anders et al. 2012; Sonnenschein and Galindo 2015). In order to enhance opportunities for all children, regardless of their family background, kindergarten needs to foster mathematics intentionally (Grüssing and Peter-Koop 2008) and children need to be provided with learning opportunities which meet their diverse educational needs (Gasteiger 2015). As with other subject areas, the quality of teaching is crucial but also highly variable (McCray and Chen 2012). Engel et al. (2016) linked the time spent on maths, as reported by the educators, with children’s maths achievement and found no correlation. They concluded that educators focus on curricular content, which is not sufficiently challenging for most children, e.g. counting and shapes.

Competencies that need to be fostered

Mathematical learning opportunities are needed which are challenging, appropriate and adaptive to the heterogeneous needs of young children. Three-to-four-year-old children need guidance to notice quantitative relationships in daily life and to begin representing ‘information about pattern, shapes, space and number’ (McCray and Chen 2012, 292). As Sarama and Clements (2009) highlight, the aim is to foster ‘overarching’ mathematical competencies, which are core for mathematics and in line with children’s thinking. Amongst these, the quantity–number–competencies are highly relevant as longitudinal studies found: quantity–number–competencies at school entry are the strongest predictor for mathematical achievement in third grade (Krajewski and Schneider 2009) and pre-school quantity–number–competencies explain up to 41% of the variance in mathematical competencies at primary school, whereas general cognitive ability only up to 10% (Dornheim 2008). The theoretical model of the development of quantity–number–competencies (Krajewski 2003) provides an orientation on the development of these mathematical competencies from the basic level ‘number word sequence isolated from quantities’ to ‘quantity to number word linkage’ to the ‘concept of number relationships’ (Krajewski and Schneider 2009, 517ff.).

Early childhood educators’ beliefs on mathematical learning

Educators’ beliefs are likely to influence the teaching of mathematics in kindergarten. Their feeling of self-efficacy regarding mathematics is related to the importance they assign to the subject in kindergarten (Brown 2005). Educators might be worried that maths is ‘not fun’ (Lee and Ginsburg 2009) and express negative feelings towards mathematics (Benz 2012), possibly shaped by their own, often negative, school experience (Anders and Rossbach 2015). Other findings indicate a positive attitude towards maths amongst early years’ educators (Chen et al. 2014; Thiel 2010). Link, Vogt, and Hauser (2017) found in a comparison of educators’ beliefs regarding fostering mathematics in kindergarten between Austria, Germany and Switzerland that the Swiss educators agree more strongly to an intentional approach to mathematics in kindergarten than the German and Austrian educators.

Different approaches for early maths

Taken that challenging, appropriate and adaptive mathematical learning opportunities are required for kindergarten and given that quantity–number–competencies are particularly relevant for later learning, educators need to decide on the best approaches to support the acquisition of these competencies in kindergarten. Schuler (2008) lays out several decisions, which educators face, amongst them the decision between (i) an instructional programme versus free learning arrangement, (ii) specific fostering of children at risk versus fostering for all children and (iii) focussing on domain-specific competencies only versus a broader approach. Traditionally, mathematics was not at the centre of curricular attention in kindergarten and educators emphasised a situated approach whereby mathematical competences are applied to everyday situations, i.e. counting the children present, comparing quantities when sharing fairly, performing one-to-one correspondence when laying the table. However, the emphasis on early learning leads to a shift in kindergarten practice. A range of learning materials are available, many of them requiring a specific time frame for focused mathematical activity, many of them designed as a training programme, with a set order of units and exercises focussing on defined skills, designed to be worked through in the given order, delivered in an educator-led instructional group setting. Educators are often not in favour of using training programmes, as they are seen as too much like school (Jörns et al. 2014). The rise in instructional, school-like training programmes for kindergarten raises the question, whether a highly educator-centred, instruction-focused approach is best suited for children of kindergarten age or whether a play-based approach would be more appropriate (Hauser 2005).

Play as an innovative approach

Innovative approaches to early mathematics should not only be developmentally adequate and effective, but also compatible with the kindergarten pedagogy. As kindergarten children are highly motivated to learn, but not in a formal, instructional way, play can be regarded as a powerful vehicle for learning (Hauser 2005). Play can be defined as activities that ‘are fun, voluntary, flexible, involve active engagement, have no extrinsic goals, involve active engagement of the child, and often have an element of make-believe’ (Weisberg, Hirsh-Pasek, and Golinkoff 2013, 105).

Play and playfulness are at the core of early childhood education (Singer 2013), although educators are not always aware of their role in fostering play (Bodrova 2008; Vu, Han, and Buell 2015). It is important to distinguish between activities, which are play-based and adult-initiated activities, which resemble school-like tasks (Bergen 2015). Weisberg et al. (2015, 10) coined educator-led educational activities disguised as play as ‘chocolate-covered broccoli’. Wood (2009) highlights the need to distinguish between the different forms on the continuum between free play and no play. Such a clarification is sought with the concept of ‘guided play’: ‘guided play sits between free play and direct instruction’ (Weisberg, Hirsh-Pasek, and Golinkoff 2013, 105) and consists of adults structuring of the play environment but leaving control to the children within the environment (Weisberg et al. 2015).

Innovative approaches to early mathematics could draw on play, be it role-play (van Oers 2010) or board and card games. It needs to be recognised that role-play, or pretend play, requires much time for the children to set up the play-frame, to engage with the play and develop it (Bergen 2015). As for board and card games, several studies found them to be effective in the acquisition of mathematical competencies (Gasteiger 2015; Jörns et al. 2014; Kamii and Kato 2005; Ramani and Siegler 2008; Schuler 2013). Gasteiger, Obersteiner, and Reiss (2015, 232) highlight that not only the concept of ‘play’ is deployed differently, but also ‘games’; consequently, they propose a ‘continuum from games designed for the purpose of entertainment only to targeted instruction with only few entertaining features’. Four aspects are essential to play-based approaches to mathematics in early childhood education: (i) the ‘mathematical content needs to be part of the mechanics of the game’; (ii) needs to be ‘correctly presented’; (iii) ‘essential for further learning’ and (iv) the game needs to be ‘appropriate for the individual learning needs of the child’ (Gasteiger, Obersteiner, and Reiss 2015, 233f).

Although play is widely acknowledged as an important learning path in early childhood education, little is known about the effectiveness of play in comparison to other ways of learning in early childhood education settings. The project presented here compared a play-based approach with card and board games with a training programme. This paper focusses on the analysis of the interviews and the learning outcomes, as the following main research question is addressed: how does the play-based approach to early mathematics with card and board games compare to the training programme regarding children’s learning outcomes and educators’ views and pedagogical preferences?

Methods

Research design

The research project compared two intervention groups, a play-based intervention and a training programme, alongside a control group in a pre-post-test quasi-experimental design based on measures of children’s mathematical competencies. The two interventions needed to be as comparable as possible regarding content and intervention time.

A training programme – known as being effective from previous testing – Mengen zählen Zahlen [Quantity, counting, numbers] was selected (Krajewski, Nieding, and Schneider 2007, 2008). It consists of 24 units of half an hour each, focussing on quantity–number–competencies. It is educator-led, addressing a small group of children, using specific tasks, maths talk and materials.¹ A play-based approach using card and board games was developed by the research team matching the curricular content of the training programme, thus it involves comparing quantities, counting, number recognition and part-and-whole relationship. Some of the games were already available, such as Halli Galli; others required an adaptation of the rules or materials, such as Shut the Box and Lining up the Fives; and some games were developed from scratch for the project, such as More is More ² (Vogt and Rechsteiner 2015). All games were piloted with kindergarten children and evaluated for their suitability in collaboration with three kindergarten educators. For the play-based intervention, the educators were provided with a box of the 10 specific card and board games. The play-based intervention was of the same duration as the training programme: All the children played the card and board games during 24 half-hour units in small groups. In general, the children could choose their co-players and a game, but they were required to play the ‘maths games’ provided in the box during the half-hour units of the intervention. The educators introduced the games and supported the children. The kindergarten educators of the control group were asked to carry on their practice as always. Widely used ways of fostering mathematical competencies include counting in day-to-day situations, playing with dice and role-play. All kindergarten teachers are required to teach the competencies defined in the curriculum but are free in their choice of pedagogical approach. Regarding research ethics, it can be stated that the research project experimentally varying pedagogical approaches with interventions of eight weeks within the framework of the kindergarten curriculum did not subject children to any disadvantage. Parents and children were informed about the research project and gave full consent to participation.

The educators of both intervention groups received the same general introduction into the learning of mathematics in kindergarten (1 day) and an introduction into either the play-based approach or the training programme (1 day) and two separate follow ups (2 half-day meetings). Data on educator’s views were collected at the end of the intervention by an independent researcher using semi-structured problem-centred telephone interviews.

Sample

From the list of all kindergartens in the Canton of St. Gall in Switzerland, kindergarten educators were contacted at random and invited to participate in the research and randomly assigned to one of the groups. In each of the participating kindergarten, the group of children in the last year before entering primary school, i.e. five-to-six-year-old children, participated. The sample included 12 kindergarten educators and 111 children in the intervention group using the training programme Mengen zählen Zahlen, 11 kindergarten educators³ and 91 children in the intervention group implementing the play-based approach and 12 kindergarten educators and 127 children as the control group. The children’s mean age was 6 years and 3 months, with no differences between the three goups (F[2,325] = 1.400; p > .05).

Instruments and data analysis

The quantitative research instruments and the statistical procedures are outlined first, followed by the qualitative instruments and the qualitative data analysis.

Quantitative instruments: the mathematical competencies were measured using the test Zahlenstark, developed by Moser and Berweger (2007) and employed for a large-scale evaluation in Switzerland (Moser and Bayer 2010). The test is compatible with the Krajewski model of the development of mathematical competency, integrating approaches of measuring mathematical competencies used by Krajewski (2003), Van den Heuvel-Panhuizen (1995) and Moser Opitz (2001). The test involves tasks on ordinality, cardinality, quantity, number knowledge and first arithmetic operations, often proceeding from tasks illustrated with images and embedded in an everyday story to numbers-only representations. Research assistants conducted the test one-to-one with each child on the premises. Cognitive abilities were measured using two subtests from CFT1 (Weiss, Cattell, and Osterland 1997). Parents completed a questionnaire with questions on the socio-economic background of the family, languages spoken at home and the home learning environment.

Quantitative data analysis: In order to ascertain whether the groups were comparable, an analysis of variance (ANOVA) on children’s age, cognitive ability, socio-economic status of the family, migration background and pretest mathematical competencies was conducted. Then the results of the mathematical competencies tests were compared by applying an analysis of variance with repeated measures.⁴ Differentiated outcomes were explored dividing the children into three groups, a third of the overall sample according to pretest mathematical competency forming a high-level group, medium-level group and low-level group, and ANOVAs were performed on each subgroup.

Qualitative instruments: The kindergarten educators of both intervention groups were interviewed by an independent researcher after the intervention. The interviews were held on the phone and had a duration of 30–40 minutes. The educators were first asked to imagine a scenario, whereby they would explain to a colleague, what the project they participated in was about. Then they were asked to describe how they implemented the intervention, how the children engaged with the intervention and – as an overall concluding evaluation – whether they are likely to use the play-based approach or the training programme in the future. The semi-structured problem-centred interviews were audiotaped and transcribed in full.

Qualitative data analysis: The transcripts were first analysed using qualitative content analysis with the software MAXQDA (Kuckartz 2010). Whilst as a first step, the interviews were carefully analysed in order to identify problems with specific games, the re-analysis of the interviews presented in this paper focusses on educators’ views and experiences. Furthermore, the response to the narrative scenario question at the beginning of each interview as well as their sense-making on pedagogical approaches to mathematics in kindergarten was analysed in detail for each educator focussing on discourse (Kruse 2015).

Results

The results to the question addressed in this paper – how the play-based approach to early mathematics compares with the training programme regarding educators’ views and pedagogical preferences and children’s learning outcomes – will be set out as follows: first, the quantitative results on children’s mathematical competencies and, then the qualitative results on educators’ views.

Results regarding children’s learning gains

The three groups proved to be comparable at pretest, as no differences were found regarding cognitive abilities, socio-economic status and migration background (details see Hauser et al. 2014). The mathematical competencies of the children at pretest do not differ between the three groups (Table 1). The treatments have an influence on mathematical learning gains, the ANOVA for repeated measure reveals a significant interaction between time and group (F[2,321] = 4.04; p = .019; partial η²= .025) (Table 1) with higher learning gain for the play-based intervention compared with the control group (Bonferroni post hoc p = .015). The calculation of the effect size results in Cohen d = 0.30, a small size effect (Cohen 1988).

Table 1. Mathematical competencies at pre- and post-test.

	M (SD)^a pretest	M (SD)^b post-test	M (SD) learning gains	ANOVA time × group	Post hoc Bonferroni
Play-based	63.89 (17.14)	75.24 (17.59)	11.35 (9.15)	p = .019;	Play > control, p = .015
Training programme	65.21 (19.78)	74.25 (17.89)	9.05 (8.70)
Control	60.58 (17.89)	68.59 (18.03)	8.02 (7.87)

^aNo difference between the groups at pretest (ANOVA: F[2,323] = 1.988, n.s.).

^bANOVA for repeated measure: F[2;321] = 4.04; p = .019; partial η²= .025.

In order to determine whether children might benefit in different ways from the intervention depending on their competency level, the children were divided into three groups according to their pretest maths competency. A third of the group was assigned to the groups of low competency level, middle level and high level, respectively. Table 2 provides the overview of competency measures for the three groups.

Table 2. Mathematical competencies of low level, middle level and high level.

Competency level at pretest	N	Mean pretest (SD)	Min pretest	Max pretest	Mean post-test (SD)	Learning gains (SD)
Low	109	43.08 (9.09)	14	54	55.33 (11.78)	12.25 (8.20)
Middle	102	62.81 (4.53)	55	70	71.44 (8.45)	8. 63 (7.85)
High	113	82.55 (10.40)	71	109	89.56 (12.68)	7.01 (8.87)

The learning gains of the groups according to level differ (ANOVA, F[2,321] = 11.416; p = .000) and post hoc Bonferroni tests show significantly higher gains for the low-level group compared to both, the middle group (p = .005) and the high group (p = .000), whereas learning gains of the middle and high group do not differ. For all three level groups, an ANOVA of repeated measure was conducted to detect possible differences according to the treatment. In the low level group, a tendency was found (F[2,106] = 2.84, p = .063, partial η² = .051) with the training programme possibly enabling a higher learning gain than the control group (Bonferroni post hoc, marginal significance, p = .078). For the middle group, there is no significant interaction regarding time and group. For the high-level group, a tendency was found (F[2,110] = 2.77; p = .067; partial η² = .048) with a marginally significant difference between the play-based approach showing higher learning gains than the training programme (p = .065) (Table 3).

Table 3. Learning gains of low level, middle level and high level according to treatment groups.

	M (SD) learning gains play	M (SD) learning gains training	M (SD) learning gains control	p time × group	Post hoc Bonferroni
Low	13.16 (8.73) n = 32	14.21 (6.99) n = 34	10.23 (8.34) n = 43	.063	Training > control p = .078
Middle	11.09 (9.11) n = 23	8.88 (8.50) n = 32	7.26 (6.48) n = 47	n.s.
High	9.82 (9.53) n = 34	5.18 (8.13) n = 44	6.57 (8.66) n = 35	.067	Play > training p = .065

Results regarding the educators’ views and integration into their pedagogy

The results from the qualitative interviews are reported as follows: first, the overall characterisation of the interventions by the educators is analysed; second, educators’ assessment on suitability and learning gains is described and third, their experiences regarding the integration of the intervention into their pedagogy is summarised.

Characterisation of the interventions: When asked how they would describe the project to a colleague, the educators of the group with the play-based approach mentioned the keywords ‘games’ and ‘mathematics’ and the organisation of the project:

I am taking part in a project on early mathematical competencies and I find it exciting. We received a whole box with materials and introduced the games and allow the children to work independently afterwards. Several competencies are fostered and it is very varied and exciting and the children are interested. (play7)

The educators of the group with the training programme often used the term ‘training programme’ (7/12) and emphasised the mathematical content (8/12):

So I would tell her [the imagined colleague] that I follow a training programme, which I use three times a week with the children … It focusses on the numbers zero to ten, in order to learn the basics, such as what does a number actually mean, or a quantity, and so to build a foundation for later arithmetic. (train11)

Educators’ assessment on suitability and learning gains: Almost all educators in both groups mentioned the quality of the material provided. The games as well as the training programme contain material made of wood, which for some of the educators is important. All educators would use the material again in the following year. Whereas almost all educators (10/11) plan to implement the play-based intervention in the following year in a similar way, only half of the educators (5/12) would implement the training programme again. Several educators emphasise that they would use the material of the training programme, but adapt the pedagogical approach and only target children with low mathematical competencies:

I would use it again but would proceed selectively. I would gather the really weak children and work more intensively with them. I would pick out some of the attractive things, for example build the number road, as an offer within free play for all children. (train3)

For the play-based approach, the educators would do the same intervention again. As this educator emphasised, all children benefitted from the intervention:

I will certainly do it again next year. I thought I would do it exactly the same way.  … The weaker children have benefited a lot, but also the strong ones. (play5)

For the training programme, all educators criticised that it did not meet the needs of all children and mentioned the problem of boredom; half of the educators said, repeatedly, that the children became very bored:

They really did not find it very cool anymore, they did not engage anymore because they were not sufficiently challenged. (train9)

Several educators also expressed their concern that the children had to sit and listen for a long time. For the play-based intervention, boredom was not mentioned at all, only one educator felt, that the children’s motivation stayed not as high as she hoped.

Integration of the intervention into their pedagogy: Several educators from the play-based approach expressed that the pedagogy of kindergarten should not entail as many programmes. While the training programme on phonological awareness is widespread and sometimes deemed compulsory, the educators would not want more training programmes in addition to the one they already have to implement. Some educators assigned to the training programme expressed that they would have wished to be in the play-based group as this would suit their pedagogy better. They expressed concern that the children had no choice of the activity in the training programme and that it was too school like:

I think now that for me, it would have been more important to be in the other group, as there was more freedom and fun things, where the children were able to choose themselves: what is cool? What am I able to do mathematically?  … This would be more what kindergarten is about.  … . Fun got lost a bit, as they had to do what they maybe did not want to do, and what they will have to do a lot in school later. (train9)

A few educators expressed that they were first concerned, that mathematics should not really be a topic in kindergarten, but then discovered that the children liked the games:

These are really games which the children like a lot.  … it fosters several areas of competencies without the children noticing, as it is all playful … I just think that mathematics should not be as present in kindergarten. (play3)

Both interventions were integrated into classroom routines, so, for example, the way of allocating games:

I have a photograph of each child … and I assigned them to the games. So their photo was next to a place and they came into the room and looked around, “where is my photo”, and then right away began to play (play1).

For both approaches, educators appreciated that there is no theme or fantasy world suggested:

I think the material [of the training programme] is beautiful and very well suited.  … it is not so over-done with knick-knacks or gimmicks, but it is very straight forward. (train7)

Discussion

The comparison of pre- and post-test, with eight weeks of intervention in between, showed a significantly higher learning outcome for the group of play-based mathematics compared to the traditional kindergarten, but no effects for the training programme. This contrasts with other evaluations where this training programme resulted in significant learning outcomes (Krajewski, Nieding, and Schneider 2008). The significant learning gains for the play-based approach as compared to the control group underlines that it is possible to obtain learning gains with an approach of guided play, using card and board games. So far, research into play-based approaches has compared effects with a control group, but not with another treatment, and found games to be effective (Gasteiger 2015; Jörns et al. 2014; Kamii and Kato 2005; Ramani and Siegler 2008). As there are no significant differences between the play-based intervention and the training programme, it can be stated that this play-based approach is at least as effective as a highly educator-led, instructional training programme. The play-based approach adheres to the idea of ‘guided play’ (Weisberg, Hirsh-Pasek, and Golinkoff 2013): free choice for children and an emphasis on peer learning, the play is guided as the card and board games are geared to specific mathematical competencies within an educator-structured learning arrangement and controlled time frame.

Whilst the training programme delivered marginally significant learning gains amongst children with low levels of competencies – compared to the control group – the training programme is not adapted to children with higher levels. It can be concluded that training programmes delivered to the whole group run the risk of having a detrimental effect, as the majority of the children are taught curricular content they already know (Engel et al. 2016) and so become bored.

The educators’ assessment of learning gains based on their day-to-day observation corresponds with the differentiated quantitative results. They also described that the training programme was mainly beneficial for children with very low competency, but that the play-based approach served all children, from low to high competency. The play-based approach was evaluated more positively, as it was considered more fun and less school like. It was, therefore, more compatible with educators’ pedagogical beliefs, similar to findings of views on the emphasis on fun (Lee and Ginsburg 2009), positive emotions (Anders and Rossbach 2015) and ‘true’ and ‘entertaining’ play (Gasteiger 2015). For both interventions, educators appreciated that the interventions did not require a specific topic or fantasy world (Schuler 2013) but that they could integrate the approach into their day-to-day routines.

Conclusion

The study effectively demonstrates that innovative approaches to early maths can be successfully based on play, particularly on card and board games, specifically developed with regard to quantity–number–competencies as defined in the curriculum. The specific potential of card and board games can be found in the opportunity of performing mathematical activities over and over again and the motivation of a peer-group setting, whereby co-players monitor and support each other (Stebler et al. 2013). The play-based approach developed here is not free play but guided play (Weisberg, Hirsh-Pasek, and Golinkoff 2013). The aim of this study was to compare the effectiveness of two different pedagogies in kindergarten aimed at the identical quantity–number–competencies. Further research might seek to establish play-based approaches to mathematics in kindergarten which target a wider range of mathematical competencies.

The results of this study highlight the importance of meeting individual children’s diverse needs (Garrote, Moser Opitz, and Ratz 2015; Gasteiger 2015). Children’s mathematical competencies in kindergarten in the year before starting school are very diverse (Dornheim 2008; Krajewski and Schneider 2009). More instructional training programmes with a ‘one size fits all’ approach fail to challenge and empower every child. However, the findings indicate that a targeted training programme for children at risk is effective together with a range of games for different competency levels for all children’s mathematical learning.

As good as learning materials can be – their educational potential can only be realised through good teaching and learning support. Also within the play-based approach, educators perform diagnosis, structure the learning arrangement, provide an impulse, a question and demonstrate strategies for solving the mathematical problems and foster discussions on mathematics amongst the children (Wullschleger and Stebler 2016). In addition, educators’ content knowledge can be influential for play-based learning support (Oppermann, Anders, and Hachfeld 2016). In the interviews, the educators expressed their pedagogical beliefs but the possible influence of such beliefs on the learning outcome could not be statistically assessed, due to sample size. Future studies might focus on the play-based approach with a much larger sample to examine educators’ competencies and beliefs.

Bringing the results together with the interviews, it becomes apparent, that all educators had a positive attitude towards the interventions. As the interventions progressed, some expressed concern about children becoming bored with the instructional setting of the training programme. The interviews clearly reveal that the educators were more enthusiastic about a play-based approach. Their positive attitude might have been a contributing factor to the learning success of the children found in this study. As there are strong traditions within the pedagogy of kindergarten, it remains important that innovative approaches to mathematics in early childhood not only prove to be effective in terms of learning gains but also that these innovations are easily integrated into the pedagogy of kindergarten. Based on the views expressed in the interviews, as well as their description of their day-to-day experience in kindergarten, a play-based approach to early mathematics certainly has great potential to become an innovation, which will be adopted readily and widely by the practitioners in the field.

Acknowledgements

The authors thank the kindergarten educators, as well as the children and their parents, for taking part in this research project.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Franziska Vogt http://orcid.org/0000-0002-2023-0431

Additional information

Funding

This research project was supported by the Swiss National Science Foundation [grant number 100014_124485].

Notes

1 The unit 2.5 of the training programme ‘Mengen, Zahlen, zählen’ (Krajewski, Nieding, and Schneider 2007, 47f) is used to illustrate the training programme. The unit uses the so-called stairs of numbers – a representation of the numbers 1–10 on wooden blocks – structured according to height, on which the number as well as the corresponding amount of points are printed. The educator places the stair of numbers in the middle of the circle of pupils and explains: ‘7 is less than 8 – pointing to the blocks of the stairs of numbers – less things belong to 7 than to 8’. Then one child after the other is asked a question like: ‘what is less, 5 or 4? Why?’ Afterwards, each child is asked to take two different blocks, to place them in front of him/her and to say ‘[number] is more/less than [number]’. The goal of the unit is to recognise the structure of numbers, that the higher number ‘is more’, contains one more, or several more things than the lower number.

2 As one example of the card and board games ‘More is More’(Vogt and Rechsteiner 2015) is described: Each card depicts three structured quantities of points in different colours, with structured representation. Altogether, the 45 cards in the game include quantities in eight different colours. The cards are distributed evenly among players. Each child places their cards as a stack face down in front of them. In the middle of the table lies one card. Children simultaneously and continuously lay open the top card and compare quantities and colours with the central card. If their card shows ‘more’ of one of the colours than the central card, they lay their card in the middle on top, thus the central card changes continuously. If their card does not depict a quantity which is more and of the same colour than the middle card, the child places that card in a new pile and uncovers the next card. When the first stack is worked through, the child takes the new pile of his/her cards and continues. The first child to discard all his/her cards is the winner. This game aims to develop competency in subitising as children are required to compare quantities very quickly without counting in order to maintain speed. The educators are advised that for this game, children at similar competency level should play together. Unlike ‘More is More’, most other card and board games in the intervention can also be played in heterogeneous groups.

3 Initially, 12 kindergartens were recruited for the play-based approach too, but one educator had to drop out on health grounds.

4 First an ANCOVA was run on the post-test result as the depending variable, including pre-test results and cognitive abilities as covariate, and group as a factor. As cognitive ability did not prove to be a significant covariate, a repeated measure ANOVA was selected as more appropriate (Field 2009).