Keywords

Challenges in Early Mathematics Learning

The early years are significant for learning mathematics (Sarama & Clements, 2008). In addition to early learning at home, learning mathematics in kindergarten is an important factor because it is the first experience of institutional learning and builds a crucial basis for future schooling (Claesens & Engel, 2013). In this context, preschool teachers can be seen as more competent other persons who support children’s early mathematical learning. This early learning takes place co-constructively in interactions. Therefore, the OECD (2017) sees the interactions between preschool teachers and children as a key variable of an early and co-constructive learning process. It seems particularly desirable and effective that in such interactions the mathematically rich and creative ideas of the children are integrated; but especially German preschool teachers are mostly uncertain of how to realize their function as early learning partners and supporters in mathematically rich and adaptive contexts. To meet this challenge, the “design patterns of mathematical situations” (Vogel, 2014, p. 232; in the following: mathematical design patterns) provide detailed descriptions for adult persons of how to realize a “mathematical situation of play and exploration” (Vogel, 2014, p. 223). The mathematical design patterns are developed in the context of the erStMaL project (early Steps in Mathematics Learning; e.g. Vogel, 2014; Acar-Bayraktar et al., 2011). They provide information about the materials used, a description of multimodal opportunities to initiate mathematical negotiations of meaning and a guide to formulate reactions that are adaptive to the children’s different mathematical ideas. In addition, they give detailed information about the mathematical content. From the researcher’s perspective on these mathematical design patterns, the question arises as to how and to what extent the use of mathematical design patterns can foster a supportive and adaptive simultaneous (inter-)acting of the preschool teachers. A fruitful theoretical and empirical point of access to this question can be achieved through the concept of responsiveness, according to Koole and Elbers (2014), and adaption in mathematics education by Robertson et al. (2015), Bishop et al. (2016, 2022) or Beck and Vogler (2021, 2022).

Responsiveness as a Construct for Analyzing Adult-Child Interactions

In order to be able to integrate children’s contributions into a co-constructive mathematical negotiation process, it is a necessary condition - if one follows the debate on “teacher noticing” (Sherin et al., 2011), for example - to “understand” or interpret children’s contributions in terms of whose mathematical content. This “understanding” can be considered a responsive act: “One must engage with another’s idea and respond to it in order to understand it—though the manner and quality of engagement can vary widely” (Bishop et al., 2016, p. 1173). From (our) interactionist perspective, responsiveness can be characterized as a coordination process between two persons, for example a child and their person of reference, in which the person of reference orientates themself toward the child’s needs in order to respond to them in a prompt and coherent way (Papoušek, 2008). For early educational contexts, interactions can be described as responsive, which is based on the children’s ideas, and in which the learners can co-determine the orientation of the process of negotiation of mathematical meaning. This means in concrete terms that the person of reference can flexibly assess when they need to expand or narrow spaces in the interaction context, when they need to instruct and when they need to allow free activities. Also in terms of mathematical (classroom) discourse, Bishop et al. (2022, p. 10) note that “responsiveness to students’ mathematical thinking is a characteristic of classroom discourse that reflects the extent to which students’ mathematical ideas are present, attended to, and taken up as the basis for instruction”. In a responsive process, for example, a teacher takes up an unanticipated or unusual mathematical idea of a child’s contribution. He or she supports the child in an adaptive way, e.g. by encouraging him or her to clarify and justify its utterance. Therefore, a high degree of responsiveness is an example of a successful scaffolding process (Vogler & Beck, 2020; Beck & Vogler, 2022). The responsiveness of parents, (educational) caregivers or (preschool) teachers is considered a key prognostic variable with regard to the social, cognitive and linguistic development of infants and children (Bornstein et al., 2008; Eshel et al., 2006). Against this background, the general question arises as to what the reality of interaction in early childhood practice looks like concerning responsive action, or how and with what kind of assistance adults in the context of kindergarten can be able to act responsively to the (mathematical) needs of the children in their everyday lives. This general question guides the research presented in this paper. This gives rise to concrete research aims and questions, which will be described, theoretically anchored and empirically examined in the following.

To pursue the questions, in our paper responsiveness is established as a construct for analyzing adult-child-interactions. On the level of interaction, responsiveness is in our paper distinguished by the fact that, in the interactive interplay of “initiation, reply and evaluation” (I-R-E, Mehan, 1979, p. 37), the teacher adapts different interactive elements to the learners’ attainment level. In this context, as Bishop et al. (2016, 2022) also hint, an adaptation can take place in subject-specific discourses – such as mathematics – in terms of (1) mathematical content and (2) interactive discursive skills. In this conversational analytical approach we adopt in our paper (following Mehan), an aim is to empirically describe the extent to which an adult’s actions, in these asymmetrical learning situations, are aligned with the assumed attainment level of a child to offer him or her the most productive participation opportunities possible through a responsive structuring of the interaction process (Beck & Vogler, 2021; Robertson et al., 2016). A high level of responsiveness generates interactional potential in a learning situation (Vogler, 2019), which opens up possibilities for interpretation and action for the children (Beck & Vogler, 2021). Both the adult’s and the children’s subsequent actions are analyzed in our observations, as the combination is necessary to reconstruct responsiveness. The other aim is to examine, what impact the use of design patterns have of the responsiveness in mathematical adult-children-interactions.

Design Patterns of Mathematical Situations

As stated above, preschool teachers are challenged by adaptively and responsively offering mathematically rich situations to children (Benz, 2012). Therefore, the starting point for our considerations is the question of whether mathematical design patterns may support preschool teachers by providing relevant information about mathematical topics and potential interactive moves for the initiated mathematical negotiations of meaning in a mathematical interaction. A “design pattern” is a standardized description system (Vogel, 2014, p. 225 f.), which makes it possible to include both the idea of mathematical content with which the children can deal, as well as the material-space arrangement and multimodal impulses in a teaching-learning environment, for supporting early co-constructive mathematical learning in various content areas. In contrast to many other recommendations for action and material, the mathematical design patterns focus primarily on the learning child, and their learning in the interactive situation. Two cornerstones of the pattern are the mathematical content and the multimodal impulses that anticipate the interactive negotiation process. The mathematical content formulated in the situation pattern is intended to support an accompanying person (B) in interacting in a mathematically rich and technically appropriate way by enabling him or her to interpret and classify the children’s utterances and actions from a professional perspective.

On this basis of the mathematical design pattern, an adult should consequently be able to react adequately to the children’s contributions. The multimodal impulses in the description pattern are intended to support the accompanying person or the professional staff in being able to act adaptively, professionally and co-constructively in the concrete interaction situation. Since the impulses consider detailed interactive and multimodal action scenarios, which in particular take up the children’s mathematics-related interpretations and attributions of meaning and ‘develop’ them as co-constructive structures, the patterns are intended to support responsive action in the interaction in terms of both content and discursive action. To gain initial insights into whether such responsiveness manifests itself in the respective mathematical design pattern, we explore the questions in our contribution, how and to what extent the use of mathematical design pattern fosters a responsive (inter-)acting in child-adult interactions conducted in kindergarten.

Data and Methodology

To address the research questions, we analyze two contrasting cases of “mathematical situations of play and exploration” from the erStMaL project (Vogel, 2014). These two situations can be seen as paradigmatic examples of different manifestations or levels of responsiveness. They can be understood as representatives of specific patterns of responsiveness.

In this project, we have investigated adult-child interactions in 12 kindergartens. 144 children aged 3–5 were observed. Therefore, we have developed mathematical situations of play and exploration, which are conducted by a guiding adult (B) with small groups of children in preschool. The accompanying persons introduced themselves intensively to the children before the play situations were carried out to create a relationship of trust.

Every play situation has its origin in one of the five mathematical domains: numbers and operations, geometry and spatial thinking, measurement, pattern and algebraic thinking or data and probability (Acar-Bayraktar et al., 2011). They are described in the design pattern mentioned above so that the guiding adult can prepare him or herself before performing the situation. The performance of the mathematical situation in preschool was videotaped and transcribed.

Interaction analysis is used to qualitatively analyse the transcribed sequences (Krummheuer, 2012), in which the statements are first interpreted individually in the order in which they occur, to then be able to understand their relationships to each other following conversation analysis (Sidnell & Stivers, 2013). This makes it possible to work out both the negotiated mathematical meaning and the characteristics of the turn-taking system. The construct of responsiveness is used as a “sensitizing concept” (Blumer, 1954, p. 7) to describe the extent to which mathematical situations create opportunities for all children to participate.

Analysis of Two Mathematical Situations of Play and Exploration

Snail Shell – Mathematical Situation Called “Ropes 01”

In our first empirical case, we analyze the mathematical situation “ropes”, in which four children between the ages of 4 and 5 and a guiding person are dealing with ropes of different lengths (short, medium, long) and colors (blue, green, red and yellow). The group tries to sort the ropes by size. Therefore, they are doing a side-by-side comparison by holding two ropes side by side. This measurement process is quite challenging because the arm span of the children is shorter than the length of the ropes.

001

René:

but I can do something with these ropes.a

002

 

holds a yellow rope of a medium length in his hand raises a long green rope

003

Marie:

holds a coiled-up blue rope in her left hand

004

Chris:

ties a short blue rope around his head

005

B:

what can you do? show it to me.

006

René:

thrusts aside a long red rope in Marie’s direction

007

 

snail shell.

008

 

gets on his knees and puts the rope on the floor in the shape of a spiral (Fig. 1)

009

Levent:

swings a green rope above his head

010

B:

a snail shell.

011

 

looks towards Marie

012

 

great. Look. can you do this too.

013

 

looks at Levent

  1. Notes: aIn the transcribed sequence actions are printed in italics, stressed words are coded in bold letters. All specialities of the spoken language (mistakes, grammar etc.) are mentioned in the translation of the transcribed sequence.

In the following 30 seconds Chris, Levent and Marie also try to make a snail shell out of the ropes. These seconds are skipped.

088

René:

so snail shell is ready.

089

B:

g r e a t. I‘m going to do one as well

090

Marie:

takes a short blue rope

091

Chris:

puts the yellow rope away and looks at B’s rope

092

B:

let’s see if my snail shell gets bigger.

093

Marie:

positions herself behind René and lets the rope hang down to the ground

094

René:

pulls a piece of the outer edge of the spiral apart (Fig. 2)

095

 

snail is built.

096

Chris:

looks at René’s snail

097

B:

great!

098

Levent:

if we got a bigger one. I will do it even bigger.

099

B:

see if this is bigger.

100

 

puts away a short red rope, which lies between him and René, and puts his spiral in front of René

101

René:

this is smaller.

While comparing the ropes, René discovers that he can lay snail shells (#001-#008), which mathematically represent Archimedean spirals (Figs. 1 and 2). In Mehan’s three steps (1979), René’s ‘spiral idea’ can be interpreted as a form of initiation that does not emanate from the person who is considered competent, but directly from the child. The guiding person (B) responds to this initiation by asking René to show the snail shells (#005) and asking Marie, Levent and Chris to lay their ropes in snail shells too (#012). B takes up the child’s terminology and encourages the children to localize the concept of the snail shell in the material. This stimulates a cooperative negotiation process in which B involves the other children (#012, #092, #099). In addition, B seems to identify the mathematical potential in René’s idea, which becomes increasingly visible as the situation progresses. Triggered by the ‘spiral idea’, an interactive space opens up for all participants, in which they can deal with the length of the ropes from an extended, geometric perspective (#092, #099). They also take a step forward in solving their problem of comparing the lengths of the ropes by replacing their impractical approach of directly comparing two uncoiled ropes with the ‘snail shell’ idea. This becomes clear from the second scene. During this process, B also places a snail shell and makes assumptions about its size (#092). He also praises René when he builds his snail (#012). Linking almost immediately to this scene, Levent also reflects on the length of a rope and the size of a snail shell (#098). Finally, René can compare the two snail shells that he and B have laid in terms of their surface area or their diameter (#101). Since he argues with the term “smaller” (and not “shorter”), the comparison in terms of the surface area seems more likely. However, it should be noted that children of this age often use the adjective “small” synonymously with “short”, which is why both possibilities of interpretation are given at this point. There is no explicit evaluation of René’s initiation by B, but a ‘polyadic evaluation’ is implicit in the activities described, in which the group implements René’s idea and uses it to compare the ropes. In addition to René and B, Levent and later Chris are also actively involved in this, while Marie participates more receptively in laying and comparing the snail shells. Thus, all of children are given the opportunity to deal with mathematically demanding concepts such as size (area), length (ropes) and even invariance in a way that was initiated by them (Beck, 2022; Vogler & Beck, 2020).

Fig. 1
A photograph captures a pair of hands of a person seated on the floor. They fold a rope into a spiral shape.

René developing snail shell

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Fig. 2
A photograph captures snail models created using colorful ropes and placed on a mat.

Created snails

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Crosswalk – Mathematical Situation Called “Wooden Sticks 01”

In the present play and exploration situation, four children (almost 4 years old) interact with the guiding person (B). First there is a free building phase in which the children place different figures (squares, triangles) and construct buildings from sticks by arranging the sticks alternately in squares (Fig. 3). After some time, B confronts the children with the concept of the mathematical design pattern and evokes the investigation of a linearly repeating pattern sequence with the two-element basic unit of green and blue sticks. Initially, this pattern is (determined) by B by alternating five sticks in the orientation green-blue-green-blue-green. Following on from this, she asks the children to continue the pattern (Fig. 4). Jonas then places a yellow stick next to the last green one (Vogler & Beck, 2020).

001

B:

Now I don’t even know which one to take (...)

002

 

Scratches her head, then takes a blue chopstick in her hand and shows the chopstick to the children

003

 

Leonard what do you think?

004

 

looks at the boy Leonard

005

Leo:

blue.

006

B:

blue?

007

 

raises the hand with the blue chopstick again

008

 

and the yellow?

009

Leo:

Dropping (unintelligible) that doesn’t belong to blue

010

B:

looks at him and whispers

011

 

why?

012

Leo:

Because (.) because otherwise, it wouldn’t change anymore.

013

B:

Then it wouldn’t change anymore? then come here, put the blue stick along.

014

 

hands Leonard the blue chopstick from her hand

015

 

Put it where it has to go. Here.

016

 

taps her finger on the carpet, next to the sequence of patterns

017

 

look, you can put it there.

018

 

points with her finger to the pattern

019

 

you have explained it to me, otherwise it wouldn’t change anymore

020

 

looks at Leonard

021

Leo:

stretches out his arm, but doesn’t place the blue double sticks next to the blue-green sequence, but on a pile of different double sticks

022

B:

There? Look. I’ll put it there again.

023

 

takes the blue stick that Leonard put on the pile and places it at a certain distance from the already existing pattern sequence

024

 

Do we have to remove it?

025

 

points to the yellow stick in the pattern sequence (Fig. 4)

026

 

what do you think? yes or no?

027

 

looks at Jonas and Leonard

028

 

what do you think?

029

 

looks at Jeremy

030

Jeremy:

Yes. Removing.

031

B:

Why?

032

Jeremy:

Because (...) (unintelligible) (...) because otherwise, that would be weird.

033

 

looks at his hands, in which he is holding a chopstick

034

B:

Otherwise it’s weird?

035

 

takes the yellow stick out of the pattern sequence

036

Jonas:

picks up a stick from the ground and breaks it with both hands

037

 

Oh

038

 

looks down at his hands holding the broken pieces

039

B:

Oh that’s bad. Put it aside.

040

 

pushes the blue stick to the green stick (...)

041

 

Look, I’ll take the blue one over here and now comes purple. Who has to come now.

042

 

rests her head on her hand looks at Jeremy, who is holding a broken stick and says out loud

043

 

Oh I think that’s nasty.

Fig. 3
A photograph captures children seated in a circle on the floor. Colorful sticks are dispersed on a mat in the center.

Children interacting with the adult

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Fig. 4
A photograph of 7 colorful sticks placed alternately on a table.

Linearly repeating pattern sequence

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In this situation, the pattern sequence with the basic unit blue-green (Fig. 3) is addressed again and again by B (from line #001), probably to draw the attention of the children to the pattern topic. She takes up Leo’s idea of placing a blue stick next in the form of a question (#006), implicitly asking for a possible reason. She also asks him to help her continue the pattern sequence (#002). B thus implicitly evaluates the boy’s answer positively. Then she asks directly for the yellow stick, which interrupts the blue-green pattern sequence (#008). Leo explains that this is not one of them (#009). This time B explicitly asks in line #011 for a reason and Leo answers that the change (between blue and green) would otherwise be interrupted (#012). B then uses Leo’s formulation and repeats his statement in the form of a question, “Then that wouldn’t change anymore?” (#013), and at the same time motivates him to put down the blue stick instead of the yellow one. At this point, Leo seems to step out of the interaction, instead placing the blue stick on top of a stack of different sticks. Then B turns first to Jonas and Leo (#027) and then to Jeremy (#029) and integrates both boys by asking them if they should put the yellow stick away (#026 and #028). Jeremy affirms this, because otherwise, “it would be weird” (#032). However, the boy only actively participates in the interaction for a brief moment before he, like Jonas, grabs a wooden stick and breaks it (#036). It becomes clear how the children actively and productively participate in the interaction and provide reasons for the statement that the yellow stick cannot belong to the pattern, but how a long-term and cooperative negotiation process fails to materialize. In addition, the polyadic interaction structure that can be reconstructed at the beginning of the situation, in which the children build and discuss their structures, ‘disintegrates’ into changing child-companion dyads in which the children, who are not actively productive, ‘flee’ and turn to other things again. Although the guiding person (B) repeatedly opens up opportunities for participation for the children and tries to tie in with the children’s contributions by adopting their formulation and terminology (lines #006, #013 and #034), it can be assumed that their persistent insistence on the two-element pattern sequence, which obviously deviates from the ideas of the children, the children gradually withdraw from the negotiation process. In addition, their strategy of repeating or taking up the children’s formulations in the form of questions (#006, #0013, #034) seems to have an unfavorable effect on the course of the interaction. The children may understand the question as implicit criticism and not as an extended request to justify or explain their statements. Breaking the chopsticks towards the end of the situation proves to be much more exciting than dealing with the pattern sequence, which can possibly also be attributed to the fact that B uses the children’s mathematically creative ideas to lay out figures and erect tall buildings, thus switching to the mathematical area of quantities and measurements is not taken up here.

Empirical Findings Concerning Responsiveness in Mathematical Children-Adult-Interactions Through the Use of Mathematical Design Patterns

In the comparison of the information from the pattern itself and the analysis of the two situations, it can be assumed that the accompanying persons in both situations are familiar with the mathematically comprehensively prepared situation pattern. This is evident from the fact that in both examples, mathematically demanding topics are negotiated with the support of the guiding adult (B), such as the invariance of the length of the ropes, which are highlighted in the mathematical design patterns. In the second example, crosswalk, this becomes clear from the fact that B, although the children build buildings and place figures, introduces the patterns’ sequence shown in the design pattern and obstinately pursues it as a theme. In the first example, the use of the mathematical design pattern obviously supports the guiding person (B) in interacting supportively by highlighting René’s idea and integrating/developing it in the group for an easy way to compare different rope sizes. B creates linguistically interactive and thematically adaptive opportunities for participation with the children and thus opens up interactional potential (Vogler, 2019) for the children. This leads to a high degree of content-related responsiveness as well as to a high degree of interactive-discursive responsiveness (Vogler & Beck, 2020; Beck & Vogler, 2021). In the second example, the use of the mathematical design pattern seems to lead the guiding person (B) into an attitude of inertia. B strictly adheres to the focus of the mathematical design pattern and tries to enforce the theme of pattern laying, even though the children are engaged in building and gaining their own mathematical experiences. The sudden change from the building phase into the pattern sequences, which is initiated by the adult, does not pick up on the mathematical ideas of the children and may generate disinterest on their part. As a consequence of this interpretation, we reconstructed a low degree of content-related responsiveness in the second example. While B persistently builds the sequence of patterns in the zebra crossing scene and seems to take over all the actions here, the children are often only passively active as silent listeners and show little interest. Despite the efforts of the accompanying person to take up the children’s moves on an interactive-discursive level, e.g. in repeating their central utterances, this results in a very one-sided negotiation. We have reconstructed a high degree of interactive-discursive responsiveness but the children are only less engaged in the mathematical negotiation process that is initiated by the accompanying person.

Conclusion and Outlook

The paper aimed to investigate empirically how the use of design patterns influence the behavior of adults in mathematical situations in kindergarten and thus lead to a high level of responsiveness in interactions. In our analyses, it could be shown how the patterns promote responsiveness: one of the prepared accompanying persons react adaptively to creative mathematical ideas of a child while working on a mathematical problem. It can be traced that the design pattern plays a major role in recognizing the child’s mathematical idea and in the subsequent interactive and co-construction-promoting implementation. Especially through the actions of the guiding person - as can be seen in the first analysis - an interactive space in the group emerges, in which the children can collectively deal with mathematically demanding concepts. In the context of the analyses, it becomes apparent, in line with the research findings of Bishop et al. (2016) and Robertson et al. (2015), how helpful the construct of responsiveness is for reconstructing the supportiveness of interaction systems.

The paradigmatic examples in this paper cannot be used to clarify how great the influence of the preparation of the guiding person is. The causal connection can also and especially be suspected through the comparative example of the second analysis: here it was possible to understand that the design patterns do not always have a positive effect on responsiveness in a concrete situation if they are interpreted as rigid guidelines for action that must be implemented in any case and in full. However, it can be assumed that this is due less to the structure of the mathematical design pattern and more to its situational implementation. In this respect, the pattern is a very promising form of preparation for mathematically rich and responsive child-adult and children-preschool teacher interactions. Due to the comprehensive mathematical description and diverse multimodal action impulses, the patterns offer a basis for being able to foster the mathematical development of children of preschool age. In our paper, we were also able to show how fruitful the construct of responsiveness is for analysing the interactive quality of early negotiation processes in kindergarten, and the assessment of supportive structures. With the help of the construct responsiveness, it was possible to work out in the context of the analyses presented here to what extent the support of the adults (accompanying persons) is responsive in terms of language, but above all also in terms of content.

Further analyses should provide detailed information about the interactional facets in which responsiveness emerges in a mathematical negotiation process between children and a guiding adult. On a theoretical level, the concept of responsiveness applies further differentiation. As the two examples show, a distinction between content-related and interactive-discursive responsiveness proves to be a viable basis for an extended, empirically-based theory genesis. Within the framework of the theoretical examination of the adaptivity of early childhood support, responsiveness can be viewed as a central, situational element of adaptive actions by elementary education professionals, which can be reconstructed in children-preschool teacher discourses and is therefore empirically identifiable and measurable.