Teachers’ Responses to Six-Year-Old Students’ Input: Learning Opportunities in Early Mathematics Education

Abstract

This study explores how teachers in Swedish preschool classes respond to and elaborate on six-year-old students’ input during mathematics teaching. The aim is to understand how different teacher responses offer different learning opportunities related to numbers and number relations. Data were collected through classroom observations of 145 teaching episodes across 95 preschool classes. Each episode was analyzed using an adapted version of the Mediating Primary Mathematics framework. Four qualitatively different response categories, predefined within the framework, were used to capture variation in how student input was responded to. Although most teaching episodes involved brief confirmations, only a few episodes included elaborations that offered students opportunities to engage more deeply with mathematical ideas. Two teaching episodes were selected for a closer analysis, using variation theory of learning, with the aim of describing the different learning opportunities offered depending on how the teachers responded to and elaborated on student input. The results show that, in both teaching episodes, students were given opportunities to learn that numbers can be represented in different ways. In one episode, the teacher’s elaboration enabled students to learn how to use units of five to count collections larger than five. In the other episode, although groups of five were represented visually, they were not explicitly used as a strategy for determining the total, and no alternative to counting by single units was offered. The study highlights the importance of how teachers respond to student input and how these responses influence what becomes possible to learn in early mathematics education.

1. Introduction

Early mathematical skills are strong predictors of children’s later learning outcomes and overall academic success (see, e.g., Aunio & Räsänen, 2015; Duncan et al., 2007; Watts et al., 2018), which highlights the importance of supporting children’s mathematical development from an early age. A key part of this development is children’s numerical understanding and their ability to recognize and manipulate numerical relationships (e.g., Sarama & Clements, 2009; Siegler & Ramani, 2008). To support this development, it is the mathematical content itself that matters but also how teachers interact with, respond to, and build on student’s thinking, a dimension that research has identified as crucial for supporting children’s mathematical understanding (Emanuelsson & Sahlström, 2008; Maunula, 2018; Murata, 2015). Student input, such as an answer or an enactment, can be responded to by teachers through questioning, feedback, and elaboration. These responses can shape the learning opportunities available and influence children’s engagement, confidence, and mathematical understanding. By actively engaging with students’ input during mathematical activities, teachers can create opportunities to deepen mathematical ideas, concepts, and reasoning, thereby facilitating learning (Murata, 2015). Thus, a teacher’s ability to ask meaningful questions, stimulate reasoning, and incorporate student input into the teaching of specific content is essential (Mason, 2000; Moyer & Milewicz, 2002; Wiliam, 2011).

Recent research has highlighted that changes in how teachers respond to and elaborate on student input can reflect broader developments in teaching practice. Askew et al. (2019), for example, show how teachers’ increased attention to students’ mathematical thinking over time was associated with more coherent and connected instruction. Such changes may reflect a shift towards more well-considered and structured teaching, in which teachers increasingly engage with and use student input to support deeper mathematical understanding and learning.

In Sweden, the preschool class is a compulsory one-year school form for all six-year-old children, serving as a bridge between preschool and primary school. Its aim is to support children’s development and learning by combining play-based approaches with more structured teaching. In mathematics education, the curriculum emphasizes foundational concepts such as counting, number sense, pattern recognition, spatial awareness, and problem solving (Swedish National Agency for Education, 2022). These concepts are typically introduced through hands-on, exploratory, and play-based activities that help concretize abstract ideas and build early mathematical reasoning. One key objective in preschool class is to nurture students’ interest in mathematics and to lay a foundation for further learning through interactive and communicative teaching that encourages children’s active participation, engagement in discussions, and collaborative problem solving. However, young students may express their ideas in non-verbal or incomplete forms, requiring teachers to interpret and scaffold their input thoughtfully. Creating an inclusive environment where all contributions are valued enables students’ ideas to become part of the learning process. This underscores the importance of teachers’ interpretative work in real-time classroom interactions (Ekdahl & Runesson, 2015), especially when aiming to support students’ learning.

Although previous research has explored early mathematics teaching, less attention has been given to how teachers respond to and elaborate on young students’ input in classroom interactions in mathematics teaching. Most existing studies have focused on older students (e.g., Emanuelsson & Sahlström, 2008; Maunula, 2018), leaving a gap in understanding how teachers in early education elaborate on young students’ input in mathematics teaching. To address this gap, the present study investigates how six-year-old students’ input is responded to and elaborated on by teachers during mathematics teaching in Swedish preschool classes and how teachers’ responses offer different learning opportunities. By combining a broad mapping of teacher responses with a detailed analysis of two selected teaching episodes, the study aims to contribute to the understanding of how different ways of responding to student input influence the learning opportunities made available, particularly in relation to how a specific mathematical content is made visible to students.

The study is guided by the following research questions:

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What qualitative differences can be identified in how teachers respond to and elaborate on student input during mathematics teaching in preschool classes?

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What different learning opportunities are offered to six-year-olds in relation to numbers, depending on how teachers elaborate on student input?

2. Literature Review

It is well established that student talk and active participation are crucial for fostering meaningful learning in mathematics, as it provides opportunities for students to engage with the content and develop deeper understanding (Franke et al., 2007). The way teachers interact with students affects their learning opportunities (Mercer & Dawes, 2014; Pianta et al., 2012), and the quality of classroom interactions plays a crucial role in facilitating student learning (Drageset, 2015). Whether initiated by teachers or students, these interactions create opportunities for the exchange of ideas, stimulate students’ thinking, and ultimately enhance their understanding of content (Gillies, 2016; Liljestrand, 2002). In mathematics education, such interactions are essential for supporting students’ mathematical understanding and problem-solving skills (Franke et al., 2007).

A common structure in classroom interaction is the Initiation-Response-Evaluation/Feedback (IRE/F) pattern, where the teacher initiates a question, the student responds, and the teacher evaluates or gives feedback (Henning et al., 2012; Howe & Abedin, 2013; Khoza, 2023; Mehan, 1979; Sinclair & Coulthard, 1975). While this structure has been criticized for reinforcing teacher control and limiting dialogue, particularly when the evaluation phase consists of brief confirmations or corrections, Wells (1993) argues that this phase, when used thoughtfully, can go beyond assessing correctness. It can support continued exploration and learning, encouraging deeper student reflection on the mathematical content. In this way, the IRE/F structure does not necessarily constrain learning opportunities, but can instead be leveraged to support mathematical reasoning, depending on how teachers respond to and elaborate on student input (Khoza, 2023; Park et al., 2020; Wells, 1993). Similarly, Emanuelsson and Sahlström (2008) show that teacher questioning plays a central role in shaping classroom communication and student participation during mathematics teaching. Questions used in teaching can serve different purposes. In particular, questions that encourage student participation in discussions focusing on mathematical content are recognized as effective tools for supporting and enhancing students’ learning (DeJarnette et al., 2020). Feedback also plays a central role in promoting higher order thinking in mathematics (Hu et al., 2021). While simple feedback may confirm correctness, it often lacks the depth needed to support deeper mathematical understanding. In contrast, feedback characterized by follow-up questions and interactive exchanges can transform classroom interactions into opportunities for reflection and support students’ understanding of underlying mathematical ideas (Hu et al., 2021).

To respond meaningfully to student thinking, teachers must not only understand the underlying mathematical concepts but also know when and how to pose questions, request explanations, and encourage students to justify their reasoning. Drageset (2015) emphasizes that a deep understanding of both mathematical content and interactive teaching strategies is crucial. Guiding, doing, and requesting are not just about helping students find correct answers, but also helping to build and deepen mathematical understanding. Liljestrand (2002) similarly highlights that classrooms can function as arenas for discussion, where teachers who actively follow up on student input foster more dynamic and dialogic learning environments.

Research shows how teachers that respond to student input can shape both the content and the roles students are expected to take in their learning (Emanuelsson, 2001; Maunula, 2018). Emanuelsson’s (2001) study on the significance of questions in the classroom reveals that teachers’ questions not only affect how students understand the content but also encourage students to take increasing responsibility for asking questions to themselves and one another. Furthermore, it is also how teachers handle students’ responses, by elaboration, clarification, or redirecting questions that encourage students to take greater responsibility for their mathematical learning. Similarly, Karlsson and Wennergren (2014) illustrate how teachers in Grade 4 mathematics classrooms used student input in various ways, including follow-up questions, reinforcement, and building upon students’ ideas to shape the direction of teaching. In a study involving older students, Maunula (2018) found that adopting a student perspective in teaching enhanced the learning opportunities, with teaching relying on both student contributions and teacher-led exploration. Furthermore, the necessary aspects of the mathematical content, in her study, linear equations, were enacted in more qualitative ways in lessons where learner contributions were explored. However, balancing attention to content with student participation can be challenging (Ryve et al., 2013). For instance, Emanuelsson & Sahlström (2008) show that while student participation can lead to a simplification of mathematical content, it also creates opportunities for students to actively shape and understand the content through interaction.

Askew et al. (2019) highlight how teachers’ use of mediational tools such as mathematical tasks, artefacts, inscriptions, and teacher talk can support students in recognizing mathematical structure and generality. Their findings show that, when teachers coordinate these tools effectively and respond to student input in ways that connect to broader mathematical ideas, they offer richer learning opportunities. Although their study does not focus explicitly on student contributions, it shows that changes in how teachers orchestrate mediational means, particularly in response to student thinking, can reflect a shift toward more coherent and connected teaching. This suggests that how teachers respond to student input in real time may play a key role in supporting deeper mathematical understanding. Teachers’ enactment should therefore go beyond merely evaluating answers, instead focusing on reasoning, integrating student input, and learning processes to support learning (Mason, 2000; Moyer & Milewicz, 2002; Murata, 2015). When constructively explored, incorrect answers and misconceptions can also serve as valuable opportunities to learn (Ekdahl & Runesson, 2015; Karlsson & Wennergren, 2014; Murata et al., 2017).

In early childhood mathematics education, insights into how teachers respond to and elaborate on student input are particularly relevant, not only in terms of supporting student participation, but also in relation to what mathematical content become available for learning. Young children often express their mathematical thinking in non-verbal or partial ways, requiring teachers to interpret and extend their input in ways that make the mathematical ideas visible (Machaba & Mangwiro, 2024; Murata, 2015). Creating inclusive environments where all contributions are valued allows student thinking to become a resource in mathematics teaching, positioning the teacher as a facilitator who connects students’ input with mathematical ideas (Machaba & Mangwiro, 2024). Recognizing how teachers respond to and elaborate on student input is therefore essential, as it may influence the learning opportunities offered in the mathematics teaching.

2.1. Theoretical Frameworks

2.1.1. Mediating Primary Mathematics

The Mediating Primary Mathematics (MPM) framework was developed to analyze and enhance the quality of primary mathematics instruction, with a particular focus on Whole Number Arithmetic (Venkat & Askew, 2018). Grounded in a sociocultural theoretical perspective, the framework conceptualizes teaching as a mediated activity in which the teacher serves as the central agent connecting mathematical objects of learning with learners. MPM was designed to identify the mediational means teachers use in classrooms and to provide a fine-grained tool for distinguishing subtle variations in how these means are employed (Askew, 2019). Drawing on Vygotskian theory, particularly the notion of number as a scientific concept, the framework emphasizes the importance of supporting learners in discerning mathematical structure and generality at the primary school level. A defining feature of MPM is its explicit focus on the nature of the mathematics made available to learners. Influenced by Arzarello’s (2006) concept of multimodal semiotic bundles, MPM broadens the notion of mediation beyond inscriptions to include embodied and material forms of communication. This broader view enables a richer understanding of how mathematical meaning is constructed in the classroom. The framework identifies key mediational means: artifacts, inscriptions, talk and gesture, tasks, and examples. These are seen as culturally and historically situated, and teachers coordinate them to support mathematical understanding and build learning connections (Venkat & Askew, 2018).

Empirically, MPM was developed through research conducted in South African primary classrooms, particularly within the Wits Maths Connect–Primary project. Analysis of video-recorded lessons revealed nuanced variations in how teachers mediated content, which informed the refinement of the framework and enabled a more detailed analysis of the characteristics of teaching quality (Askew et al., 2019).

2.1.2. Variation Theory of Learning

According to variation theory, learning always has an object, something to be learned, a specific content, or a capability (Runesson, 2005). Learning involves experiencing or understanding the content in a new and different way. For this to happen, the learner needs to discern new aspects of what is to be learned (the content) (Marton, 2015). These aspects must be discerned as dimensions of variation, meaning that something varies within the aspect in focus while other things remain invariant.

The object of learning is central in variation theory because it describes what (e.g., a specific mathematical content) is to be learned by the students (Marton, 2015). A distinction can be made between the intended object of learning (the planned learning intention for a lesson), the enacted object of learning (observations made during the lesson), and the lived object of learning (what the students actually learned) (Häggström, 2008). This study focuses on the enacted object of learning, analyzing how the mathematical content was dealt with in the learning situation. While variation theory asserts that a learner can discern an aspect if it is opened up as a dimension of variation, the way the teacher emphasizes these aspects is crucial. Thus, the aspects opened up as dimensions of variation constitute the enacted object of learning. When analyzing the enacted object of learning, different learning opportunities can be identified depending on which dimensions of variation are opened up. For example, Ekdahl’s (2020) analysis of nine teachers’ enactment of the same number activity with 5-year-olds identified four different enacted objects of learning, each with a set of opened-up dimensions of variation. This resulted in different learning opportunities being offered in these learning situations.

The enacted object of learning, which is constituted in the learning situation, also depends on the interaction between teachers and students (Lo, 2012). This interaction includes what the teacher introduces, the student’s inputs, and how the teacher responds to these inputs, potentially offering different learning opportunities. Therefore, the way teachers’ respond to and elaborate on students’ input can affect which aspects of the (mathematical) content are opened up in the learning situation (Emanuelsson & Sahlström, 2008; Maunula, 2018; Ryve et al., 2013).

3. Materials and Methods

3.1. Context

This study was conducted within the larger research project SATSA (Structural Approach in Teaching as foundation for Sustainable Arithmetic learning), funded by the Swedish Research Council (grant number 2020-03712). The project focuses on mathematics teaching and learning in Swedish preschool classes, with particular attention to the quality of early year mathematics education related to numbers and number relations.

Consistent with a substantial body of research, we assume that arithmetic competence relies on children’s ability to handle numbers as structured in part–whole relations (Baroody & Purpura, 2017; Björklund et al., 2021a; Kullberg et al., 2020; Venkat et al., 2019). A structural approach to early number teaching emphasizes students’ composition and decomposition of numbers into parts and wholes, rather than simply operating on individual units. This approach supports deeper understanding of numerical structure and arithmetic (Björklund et al., 2021a, 2021b; Kullberg et al., 2024) and provides the foundation for the present study. Specifically, in this study we examine how teachers’ responses to and elaborations on students’ input in early mathematics teaching affect what aspects of number and number relations become visible to students. By analyzing how teacher responses contribute to making number structure discernible, such as through highlighting part–whole relationships or efficient grouping, this study offers insights into how a structural approach can be enacted in classrooms.

3.2. Participants and Data Collection

Classroom observations were carried out in 95 preschool classes across 56 schools, located in both urban and rural areas in ten municipalities in southern Sweden. The study was conducted with permission from school principals, and informed consent was obtained from all participating teachers. Initially, 105 preschool class teachers agreed to participate, although 10 later withdrew for various reasons. Each participating teacher independently selected a mathematics lesson for observation, with the aim of supporting students’ understanding of numbers and number concepts, an area emphasized in the Swedish national curriculum for the preschool class (Swedish National Agency for Education, 2022). The number of participating children during the lessons varied from 4 to 28.

All observations were conducted by five researchers, including the authors of this article. The lessons were documented through detailed fieldnotes taken with paper and pencil, with supplementary photographs taken on some occasions. Particular attention was paid to student input, the use of artifacts and notations, as well as to the teacher’s gestures and verbal expressions in handling methods and mathematical relationships. To structure the analysis, the lessons were segmented into teaching episodes, defined as a segment of a mathematics lesson centered around a specific mathematical activity or content. A new episode begins when there is a shift in focus initiated by the teacher, for example, by saying “Now let’s move to something else”. Each observed lesson was therefore segmented into one or more teaching episodes. In total, this resulted in a dataset of 145 teaching episodes, which provides the empirical material for analyzing how teachers’ respond to and elaborate on students’ input in early mathematics teaching.

3.3. Analysis

The analysis was conducted in two steps. In step 1, all 145 observed teaching episodes were coded using an adapted version of the Mediating Primary Mathematics (MPM) framework (Venkat & Askew, 2018), focusing on how teachers responded to and elaborated on 6-year-old students’ input in their mathematics teaching. In step 2, two teaching episodes with similar mathematical content, observed by the authors of this paper, were selected for closer analysis based on their potential to illustrate learning opportunities offered through how teachers responded to and elaborated on students’ input. This analysis was informed by variation theory (Marton, 2015):

• Step 1: Coding teaching episodes using the MPM framework.

The MPM framework (Venkat & Askew, 2018) provides a structured tool for analyzing how mathematical meaning is mediated through teaching, with analytical focus on (I) the use of artifacts, (II) the use of mathematical notations, and (III) the teacher’s gestures and verbal expressions when dealing with (a) methods, (b) mathematical relationships, and (c) student input. This article focuses exclusively on aspect IIIc: the teacher’s use of gestures and verbal expressions in response to student input. However, when applying the original MPM framework, we found that aspect IIIc, primarily concerned with teacher responses, did not fully capture the pedagogical priorities of the Swedish preschool class context. To better align the framework with the communicative and reasoning-oriented goals of the Swedish preschool class curriculum (Swedish National Agency for Education, 2022), we adapted aspect IIIc and renamed it Building learning connections: explanations and evaluations of errors/for efficiency/with rationales. Support for student input and mathematical reasoning (see

Table 1. Categories of teacher responses to student input for Aspect IIIc in the adapted MPM framework. The table presents four qualitatively distinct ways in which teachers respond to student input.

IIIc. Building learning connections: explanations and evaluations of errors/for efficiency/with rationales. Support for student input and mathematical reasoning.
Pulls back to naïve methods or no evaluation of input (correct or incorrect)	Confirms/accepts strategies or inputs; Notes/questions incorrect input	Explains student answers/inputs a more sophisticated strategy/addresses errors through elaboration	Elaborates on, explains answers, clarifies, and justifies strategy choices based on appropriateness
A	B	C	D

). This adjustment places greater emphasis on how teachers acknowledge, confirm, elaborate on, and evaluate students’ input, with attention to the relevance and appropriateness of their responses. The adaptation reflects the curriculum’s aim to provide students with opportunities to use mathematical concepts and reasoning to communicate and solve problems (Swedish National Agency for Education, 2022, p. 23).

Each teaching episode was categorized into one of four qualitative categories (A–D), describing how the teacher builds learning connections by responding to and elaborating on student input (see Table 1). The categorization focuses on the extent to which teachers provide explanations, evaluate errors, promote efficiency, and offer rationales that support students’ mathematical learning. In Category A the teacher pulls back to naïve methods or makes no evaluation of the student’s input, where, in Category B, the teacher confirms and accepts students’ strategies without further elaboration. In Category C, the teacher explains or offers a more sophisticated strategy or addresses errors through elaboration, while Category D represents a high degree of responsiveness and support, characterized by teacher elaboration and deepening of students’ reasoning. The analysis is grounded in a qualitative categorization of episodes based on teacher responses to student input. It is important to emphasize that the Categories A–D represent qualitatively different ways in which teachers respond to student input. Each category reflects specific features of teacher responses, where higher categories indicate more elaborated responses that may offer greater support for students understanding of mathematical content, rather than simply being “better” or “worse”.

To ensure consistency and shared understanding in the coding process, the research team conducted joint calibration sessions. Prior to the analysis, we reviewed and discussed video-recorded mathematics lessons from previous research projects and national professional development programs. This process allowed us to test and refine the adapted coding framework, particularly in relation to the Swedish preschool class context.

During the analysis of our own observational data from preschool class, the research team continued to calibrate interpretations by jointly reviewing and discussing specific teaching episodes. Disagreements were carefully examined and resolved through collaborative discussion until consensus was reached, ensuring consistency and validity in how episodes were coded. This collaborative approach strengthened inter-rater reliability and supported a nuanced understanding of teacher–student interactions.

To provide an overview of the distribution of teaching episodes, the frequency of each category was compiled. In cases where multiple response qualities were identified within a single episode, the highest-ranking category was used in the quantitative summary. The distribution of episodes across these four categories is presented in the results section (see Section 4.1).

• Step 2: Analyzing learning opportunities in teachers’ responses to students’ input.

While the MPM-based coding provided an overview of the qualitative differences in how teachers responded to student inputs across the 145 observed episodes, this type of categorization provides limited insight into the specific learning opportunities offered to students. To explore this further, two teaching episodes were selected by the authors of this article for a more fine-grained analysis. These episodes were chosen to illustrate what aspects of numbers were made visible through teachers’ responses to and elaborations on student input in mathematics teaching and the different learning opportunities of numbers offered within the teaching. In both episodes (Teaching Episodes A and B), the teaching centered on counting and determining quantities by clustering tally marks and sticks in groups of fives (using the five-fence symbol). The two episodes analyzed in this article serve as illustrative examples of the qualitative categories in the MPM framework (Elofsson et al., 2024; Venkat & Askew, 2018), demonstrating how teachers’ elaborations on student input offer opportunities for students to discern the mathematical concepts in focus.

The analysis in Step 2 was guided by the variation theory of learning (Marton, 2015). We examined two selected teaching episodes to explore how teachers responded to and elaborated on student input and what learning opportunities were offered to the six-year-olds. This approach enabled us to reveal aspects not visible through the initial coding. To analyze how teachers highlighted the mathematical content in their responses and the learning opportunities offered in these episodes, we used principles from variation theory (e.g., Maunula, 2018; Björklund et al., 2021b).

Drawing on fieldnotes and photographs from the observed teaching episodes, we focused on how the teachers responded to and elaborated on student inputs. Our aim was to identify which aspects of numbers (dimensions of variation) were made visible through the teacher’s elaboration and whether and how these dimensions were opened up in the teacher’s enactment. When an aspect was opened up as a dimension of variation, we identified what varied and what was kept invariant. For example, in one episode, a student placed five sticks in a horizontal line on the floor. The teacher responded by clustering the sticks into a group of five, placing the fifth stick diagonally across the four sticks saying, “like a fence”. The number (5) was kept invariant, and the way of organizing the sticks varied (either as single units or as a composed group). This opened up the aspect, five as composed units, as a dimension of variation. Through repeated analysis of the fieldnotes, we identified the dimensions of variation that were opened up in each teaching episode. Some aspects were identified in both episodes, while others were identified in one episode. The aspects that were opened up as dimensions of variation in the two teaching episodes (A and B) and thus constitute two distinct enacted objects of learning. In line with the variation theoretical assumption that discernment requires experiencing variation (Marton, 2015), this analysis allowed us to describe the enacted objects of learning and the different learning opportunities offered to students in the two episodes.

Taken together, the two-step analysis combined breadth and depth by capturing both the overall patterns in how teachers responded to and elaborated on students’ input and the specific ways in which this affected the learning opportunities available to six-year-olds. This approach allowed us to examine not only how student input was acknowledged and used in early mathematics teaching, but also how aspects of mathematical content were made discernible through teachers’ responses and elaborations, thereby affecting what becomes possible to learn.

3.4. Ethical Considerations

This research was conducted in accordance with the ethical guidelines established by the Swedish Research Council (2024). Prior to data collection, school principals and teachers received written information about the purpose of the study and were informed that participation was voluntary. Informed consent was obtained from all participating teachers. All collected were handled confidentially and stored in accordance with the University of Gothenburg’s research data handling guidelines. To ensure anonymity, neither individual teachers nor schools are named in the publication, and the data material are used solely for research purposes.

4. Results

The results are presented in three parts. First, we provide an overview of the qualitative differences in teacher responses to six-year-old students’ input across all observed teaching episodes, based on our coding using the adapted MPM framework (Section 4.1). Second, we present two teaching episodes in detail, highlighting how teachers initiated, responded to, and elaborated on student input (Section 4.2). Third, we discuss the different learning opportunities offered in the two teaching episodes (Section 4.3).

4.1. Qualitative Differences in Teacher Responses to Student Input

The analysis revealed qualitative differences in how preschool class teachers responded to and elaborated on six-year-old students’ input during mathematics teaching focused on number and number relations.

In 8.3% of the observed teaching episodes (Category A), teachers pull back to naïve strategies or provided no evaluation of the students’ input. For example, in one episode, a student was asked to add two groups of objects (4 + 3) and gave the correct answer of seven. Instead of asking how the student arrived at the answer, the teacher asked the student to count the objects one by one. This response overlooked the opportunity to build on the student’s strategy and to explore and contrast alternative approaches. In some cases, no feedback was given at all, regardless of whether the student input was correct or incorrect. For example, in one teaching episode students were asked to place cards with numbers 1–10 in order. When the numbers 10 and 6 were placed upside down, the teacher made no comment or correction, thereby missing a chance to address misconceptions.

In 61.4% of the episodes (Category B), teachers responded to student input with brief, general feedback such as “Good”, “That’s right”, or confirming by giving an approving nod. Although the input from the students was acknowledged, it was not further elaborated on. The teaching often stopped at the student’s answer, without further exploration of strategies or underlying mathematical ideas. For example, when a student was asked to represent the number six using their fingers, the teacher confirmed the answer with a brief “Good” but did not further explore the student’s strategy or reasoning. The focus appeared to be on reaching a correct answer rather than on deepening the understanding of mathematical content.

In 29.7% of the teaching episodes (Category C), teachers actively incorporated student input into the teaching by elaborating on or verifying it. This included repeating and extending student input, offering contrasting examples, or linking the input to prior content. For instance, a teacher might repeat a student’s contribution and extend it by introducing a more sophisticated strategy and/or connecting it to previous teaching content, thereby making the student input a meaningful part of the learning process. For example, in one episode, students were asked to show the number eight using their fingers. The students used different combinations: some held up four fingers on each hand (4 + 4), while others used five fingers on one hand and three on the other (5 + 3). The teacher highlighted the different representations and initiated a discussion about which representation made it easiest to “see eight” without having to count each finger individually. To further challenge the students’ thinking, the teacher also introduced a non-example (five fingers on one hand and four on the other) and asked whether that also made eight. This example illustrates how the teacher used student input to make part–whole relations visible and to support students in discerning mathematical structure through comparison and reflection.

Only one episode (0.7%) was categorized and coded as Category D, where the teacher not only built on the student input but also carefully explained and justified the mathematical reasoning involved. In this case, the teacher explicitly clarified why a particular strategy was effective, thereby supporting students’ learning and mathematical understanding. This teaching episode is presented in detail in Section 4.2 (Teaching Episode B), to illustrate how teacher responses to and elaborations on student input may influence the learning opportunities made available to students.

In summary, the analysis using the adapted MPM framework showed qualitative differences in how teachers responded to and elaborated on student input. While student contributions were frequently acknowledged by the teachers, most responses were brief confirmations that offered limited support for mathematical reasoning.

4.2. Illustration of Teacher Responses to Student Input—Two Teaching Episodes

First, we present each of the two teaching episodes individually, detailing the teacher’s initiations, students’ inputs, and the teacher’s elaborations in making aspects of numbers visible. A teacher’s initiation could be a question, a prompt, or an action directed to the students, followed by an individual student’s or group of students’ input (an answer or an enactment) which the teacher then responds to and/or elaborates on through talk, gestures, and/or notations. At the end of each teaching episode, we also provide a summary of the MPM coding.

4.2.1. Teaching Episode A

This is the first lesson of the day. The teacher and twelve students are seated in a circle on the floor. To begin, the teacher hands a box with orange wooden sticks to one of the students.

Transcript: Teaching episode A
Teacher initiates	T:	Put the same number of sticks on the floor as there are children in the circle today.
Student’s input 1	S:	Ann, Bob, Cira, Dave … For each name she says, she places an orange stick on the floor in a row in front of her.
Teacher’s response 1	T:	When the student is about to place the fifth stick on the floor, the teacher interrupts and shows how the sticks can be clustered into a group of five. Like a fence.
Student’s input 2		The student places the fifth stick diagonally over the four sticks and continues putting sticks on the floor. She ends up with twelve sticks, clustered in two groups of five and two single sticks in front of her (Figure 1).
Teacher’s response 2	T:	Count the number of sticks that are visible on the floor.

After the student pairs the names of four classmates with the sticks (Student’s input 1), the teacher responds by interrupting and guiding both the individual student and the class towards clustering the sticks into a group of five, referring to it as a “fence” (Teacher’s response 1). She offers a variation of how the same number (5) can be represented both as single units and as a grouped set of five, by highlighting the concept of five as a composed unit and introducing the five-fence symbol. After following the teacher’s instructions, the student organizes the ten sticks into two groups of five and sets two additional sticks aside (Student’s input 2). The teacher responded by asking the students to count the sticks (12) on the floor (Teacher’s response 2).

Transcript: Teaching episode A continues
Student’s input 3	S:	One, two, three, four, five, six, seven, eight, nine, ten, eleven. Pointing at the sticks, without coordinating the number words with the individual sticks.
Teacher’s response 3	T:	Let’s count the sticks together once again. Pointing at one stick at a time when determining the total.
Students’ input 4	S:	One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.

When the student counts the sticks (Student’s input 3), the teacher accepts the chosen method; however, she notices that the student’s count is incorrect (11 instead of 12). To address this, the teacher responds by offering the same counting method, saying, “Let’s count the sticks together once again”, while using gestures pointing at one stick at a time (Teacher’s response 3).

Transcript: Teaching episode A continues
Teacher initiates		The teacher asks the students about the absence of classmates today.
	T:	How many are missing from the class today?
Students’ input 5		The students discuss which classmates are missing. They agreed that five classmates were missing today.
Teacher’s response 5		The teacher places five more sticks arranged as a five-fence representing the number of missing students above the twelve sticks representing the present students. With support from the teacher …
Student’s input 6		… the student picks two number cards (5 and 12) corresponding to the number of sticks in the two rows and places one card beside each row (Figure 2).
Teacher’s response 6	T:	Now, let’s count together.
Students’ input 7	S:	Thirteen, fourteen, fifteen, sixteen, seventeen (counting on from 12)

When the students, on the initiative of the teacher, were asked to find out the number of missing classmates and agreed on five (Students’ input 5), the teacher elaborated on the input by putting five sticks on the floor, representing the absent classmates (Teacher’s response 5). Thereafter, she offered a student to pick up the cards with symbols for representing 5 and 12 and to count the five sticks that represent the absent students, starting from 12 and counting up to 17, determining the total number of students in the class (Teacher’s response 5 and 6). Thereby, it becomes possible for the students to discern that numbers, in this case numbers of classmates, can be represented with wooden sticks grouped in five, by symbols (5 and 12), and by oral counting.

Teaching Episode A was coded as Category B in the adapted MPM framework (see Table 1). Although two distinct qualities of teacher responses were identified in the episode, corresponding to Categories A and B, the episode was coded as B, in line with the coding principle of assigning the highest observed category within an episode.

In this episode, the teacher responded by drawing students’ attention to a more sophisticated strategy, by grouping the wooden sticks in fives (Category B). However, she then pulled back to a more naïve method (single-unit counting) without drawing students’ attention to the advantage of grouping the sticks (Category A) for a more sophisticated counting strategy.

4.2.2. Teaching Episode B

In Teaching Episode B, observed in a different preschool class at another school, 16 students are seated in a semicircle in front of the whiteboard as the teacher initiated the discussion.

Transcript: Teaching episode B
Teacher initiates	T:	If we are going to count points when we play games and want to write five, how can we do that?
Student’s input 1	S:	With lines.
Teacher’s response 1	T:	Okay, like this? Draws five lines on the whiteboard. (Figure 3a). It is quite difficult to see how many lines there are. Look! The teacher draws more lines in a new row on the board (Figure 3b).

The first student proposes that they could draw lines (Student’s input 1). The teacher acknowledges the student’s input and proceeds to draw five tally marks in a row on the board (Teacher’s response 1). To further clarify her point, she adds another row of eleven tally marks, commenting on how challenging it is to discern the total number of tally marks. This (Teacher’s response 1) demonstrates the teacher’s effort to extend the discussion beyond the current context by incorporating the students’ potential prior experiences. By simultaneously presenting the contrasting groups (5 and 11 tally marks on the board, as shown in Figure 3a,b), she elaborates on inputs and emphasizes the significance of grouping/clustering individual objects.

Transcript: Teaching episode B continues
Teacher initiates	T:	Have any of you seen any other way to write five using lines?
Student’s input 2	S:	A line over!
Teacher’s response 2	T:	Yes! Show us on the board!
Student’s input 3		The student draws five lines and a line diagonally across these five lines (Figure 4a).
Teacher’s response 3	T:	You are on the right track. Let’s count together: one, two, three, four, five, six, pointing simultaneously on each line.
Student’s input 4	S:	One must go.

A student suggested that there should be “a line over” the vertical lines (Student’s input 2) and was encouraged by the teacher to demonstrate on the board (Teacher’s response 2). The student draws five vertical lines with a diagonal line crossing them (see Figure 4a). The teacher acknowledges the student’s effort, even though it was incorrect, by confirming that he is on the right track (Teacher’s response 3). Then, in the teacher’s elaboration, they count the lines together while the teacher points at one line at a time. One student realizes there are six lines and that one line needs to be removed (see Student’s input 4).

Transcript: Teaching episode B continues
Teacher’s response 4		The teacher suggests that they should delete one of the lines and then count together again. She erases one of the vertical lines (Figure 4b) and counts.
	T:	One, two, three, four, five, pointing at one line at a time simultaneously.
Student’s input 5	S:	Now it’s five!
Teacher’s response 5	T:	Yes. This line is the fifth line, pointing at the diagonal line. When it is like this, circling the five-fence with her finger, it is always five, holding up five fingers. It helps us when we for example count points. We can use the five-fences and skip count in fives “five, ten, fifteen, twenty, and so on”.

When the student observes that one line needs to be removed (Student’s input 4), the teacher erases one of the vertical lines, and they count the lines together once again, concluding that they now have five lines (Teacher response 4). Building on the student’s input 3, the teacher contrasted the representation of six (comprising five units and a diagonal line) with five (comprising four units and a diagonal line). This comparison allowed the students to discern the five as a unit referred to as a “five-fence”. In this way, a variation between the intended symbol (the five-fence) and the numbers represented was offered.

The teacher uses gestures to draw the learners’ attention towards the diagonal line to emphasize how the five-fence symbol is constructed (Teacher’s response 5). The teacher further explained that they have now created a unit (a group) of five, simultaneously demonstrating this by circling the group of five and showing five fingers on one hand. This teacher’s elaboration on the student’s input demonstrates that numbers can be represented in different ways, opening up a dimension of variation where the same quantity (5) is represented through different representations (five-fence and fingers). Additionally, the teacher emphasizes that five-fences is a practical and useful method for documenting points. How these units can be used for counting in multiples of five within a higher number range, instead of relying on single-unit counting, is also emphasized in the teacher’s response (Teacher’s response 5).

Teaching Episode B was coded as Category D in the adapted MPM framework (see Table 1). Although two qualities of teacher responses were identified, corresponding to Categories C and D, the episode was coded as D, following the coding principle of assigning the highest observed category. The teacher not only elaborated on student input by contrasting correct and incorrect responses and offering more sophisticated strategies (Category C) but also explained and justified the underlying mathematical reasoning behind the five-fence symbol (Category D).

4.3. Different Learning Opportunities

The analysis suggests that different learning opportunities were identified depending on which dimensions of variation were opened up through the teachers’ initiations, responses, and elaborations on students’ input. According to variation theory, the enacted object of learning is constituted through what aspects of the content are made discernible to students in the learning situation. In this study, the aspects that were opened up as dimensions of variation in teaching episodes A and B constitute two different enacted objects of learning, referred to here as enacted object of learning A and B, respectively.

In both enacted objects of learning A and B, students were given the opportunity to learn that the same numbers can be represented in various ways. In enacted object of learning A, variation was offered through the teacher’s elaboration, offering different representations of numbers, such as manipulatives, numerals, and number words. In enacted object of learning B, variation was offered through written lines on the board, fingers, and number words.

Differences in learning opportunities were related to how teachers responded to and elaborated on students’ input. In enacted object of learning B, students’ inputs to the teacher’s initiation focused on strategies for recording points and using units of five as an efficient method for calculating larger quantities in scoring games. Through this elaboration, students were given the opportunity to understand not only how the five-fence represents a group of five, but also why this representation is efficient and useful for counting larger quantities. By acknowledging, verifying, extending, and advancing students’ inputs, while also addressing misconceptions and providing non-examples, the teacher guided the lesson in a purposeful and distinct direction. This kind of elaboration was not identified in enacted object of learning A.

In the enacted object of learning B, the students were given the opportunity to learn how units of five, and specifically the five-fence symbol, can be constructed and used as composed units for counting a collection larger than five. Variation was offered when the teacher drew different quantities of tally marks (5 and 11), side by side, on the board, enabling students to discern them simultaneously (Figure 3a,b). Further contrast was presented (see Figure 4a,b) between the representations of six (five vertical lines plus a diagonal) and five (four vertical lines plus a diagonal). These contrasts made it possible for students to understand that a long sequence of tally marks can be difficult to track when representing larger quantities and to discern both the efficiency and characteristics of the five-fence symbol. In the enacted object of learning A, the teacher’s elaboration on student input offered alternative learning opportunities. Teacher A accepted both the students’ strategy (single-unit counting) and noted incorrect answers (e.g., identifying 11 instead of 12), responding by encouraging students to recount the wooden sticks (the number of items varied; the strategy was the same). The results suggest that students had the opportunity to learn how to determine quantities greater than five through single-unit counting.

Like Teacher B, Teacher A introduced the five-fence symbol. However, Teacher A did not elaborate further on the students’ input to explain why grouping in fives is efficient or the significance of placing the fifth stick diagonally across the four others. As a result, a variation of this aspect was not offered. Furthermore, when determining how many students were present or absent, Teacher A did not offer alternative strategies beyond counting by ones, limiting the potential for students to explore more efficient methods. Although the five-fence symbol was made visible through orange sticks grouped in fives and placed on the floor throughout the episode (see Figure 1 and Figure 2), the teacher did not explicitly elaborate on students’ input or offer alternative counting strategies. While students could visually discern how to cluster objects into groups of five and how the five-fence symbol can be used as a composite unit for counting, this was not underpinned by the teacher. For instance, although the visible arrangement (three groups of five and two additional sticks) could have supported counting by fives (five, ten, fifteen, sixteen, and seventeen), this strategy was not introduced or elaborated upon. Thus, without further elaboration or offering variation in strategies for determining quantities greater than five, the learning opportunities may have been more limited, compared to what was offered in Teaching Episode B.

5. Discussion

This study examined how teachers in Swedish preschool classes responded to and elaborated on six-year-old students’ input during mathematics teaching and how these responses offer different learning opportunities. Using an adapted MPM framework, we coded 145 teaching episodes, identifying qualitative differences in how student input was acknowledged, elaborated on, or left unexplored. To gain a deeper understanding of the learning opportunities offered through teachers’ responses, two episodes were analyzed in depth using a variation theory perspective, which allowed us to examine how dimensions of variation were opened through the teachers’ responses and how these responses contributed to constituting different enacted objects of learning.

The coding of the teaching episodes revealed that student input was most often acknowledged through brief confirmations (Category B), such as “Good” or “That’s right,” without further elaboration from the teacher. This reflects a typical IRE/F structure, where evaluation ends the interaction. These brief responses rarely led to further exploration of students’ strategies or reasoning, thus limiting opportunities to make key mathematical ideas visible (cf. Hu et al., 2021; Wells, 1993). In some episodes (Category A), teachers even pulled back to more naïve strategies, such as single-unit counting, despite students demonstrating more advanced methods. This limited the opportunity to highlight variation in strategies and to make aspects of the mathematical content visible. This is particularly relevant considering the study’s overall aim to understand how teacher responses affect the learning opportunities offered, especially in relation to number and number relations, key areas in early mathematics development (Sarama & Clements, 2009; Siegler & Ramani, 2008). This finding supports previous studies showing that, when teachers do not actively engage with and build on students’ input, opportunities for reasoning and deeper understanding of mathematical content may be missed (Emanuelsson, 2001; Murata, 2015). While Ryve et al. (2013) point to the risk of content simplification when emphasizing student participation, in our data simplification occurred not due to student participation, but could instead be related to the teacher’s limited elaboration on the students’ input.

In contrast, Category C episodes illustrated how teachers responded to student input in a more qualitative way allowing students to notice patterns, compare strategies, and build on their prior knowledge. Teaching that actively builds on student input in this way reflects what Askew et al. (2019) describe as an effective use of mediational tools where teacher talk and student input support recognition of mathematical structure and generality. This approach aligns with the Swedish curriculum’s emphasis on interactive and communicative teaching in preschool class, where students are encouraged to participate actively and explore mathematical ideas (Swedish National Agency for Education, 2022). Notably, only one of the episodes (Teaching Episode B) out of 145 was coded as Category D. Although rare, it illustrates how elaboration can offer learning opportunities that highlight mathematical structure and supports understanding of the mathematical content. These findings raise questions about how teachers can be supported to move beyond brief confirmations and instead engage more deeply with (young) student contributions, whether correct or incorrect, in ways that promote reasoning, discussion, and understanding of mathematical ideas. As Wells (1993) and Khoza (2023) notes, the evaluation phase can open further exploration if used thoughtfully. Similarly, Emanuelsson and Sahlström (2008) emphasize the importance of teacher follow-up in shaping student participation and access to content, which was visible in approximately one-third of the teaching episodes in this study. At the same time, it is important to acknowledge that not every student input requires an elaborated response (as in Category C and D). In some cases, a brief confirmation, a gesture, or a short verbal response may be sufficient. However, when student input offer possibilities to deepen understanding or clarify mathematical ideas, the absence of response or elaboration may represent a missed opportunity to learn. The challenge for teachers lies in recognizing (in real time) when elaboration can meaningfully contribute to learning and to respond in ways that balance pedagogical appropriateness with the potential to make mathematical content visible. Thus, the aim is not to suggest that all teaching episodes should be categorized as C or D, but rather to highlight the importance of responsiveness and professional judgment in identifying when and how to build on student input in ways that support mathematical learning.

In this study, two complementary analyses were conducted to address the research questions. The first analysis, based on the MPM framework (Venkat & Askew, 2018), provided an overview of the qualitative differences in how student input was taken as points of departure. The second analysis, grounded in variation theory (Marton, 2015), offered a more detailed description of how mathematical content was enacted through variation, with student input serving as a starting point. This theoretical approach allowed us to examine how aspects of the content were made discernible to students through patterns of variation and invariance. The findings suggest that teaching episodes coded as Categories C and D within the adapted MPM framework were more likely to open up dimensions of variation, thereby enhancing the potential for student discernment. For example, in enacted object of learning B, the coding of teacher response predominantly reflected Categories C and D, indicating richer learning opportunities. In contrast, enacted object of learning A was coded as Categories A and B suggesting more limited opportunities for learning. However, the nature of the mathematical content in focus and the teacher’s initiation may also influence the students’ learning opportunities. In this study, Teacher B initiated the episode by asking how to score points in a game, while Teacher A initiated by asking students to count classmates. These different initiations may have constrained the potential for deeper mathematical engagement, depending on how the tasks were framed and elaborated on by the teacher.

In enacted object of learning A, although the five-fence symbol was introduced, its rationale or its potential for supporting efficient counting was not made explicit. Although the wooden sticks are present, the teacher does not explicitly guide students to reflect on the relationships between the single-unit counting and the benefits of grouping in fives. In contrast, in enacted object of learning B, the teacher not only builds on students’ inputs but also offers contrasts between correct and incorrect answers, explains the rationale behind the five-fence symbol, and connects it to efficient counting strategies. This aligns with the principles of variation theory (Marton, 2015), emphasizing that learning occurs when learners discern aspects of content through variation. By highlighting differences between five and six (Teacher response 3 in Teaching Episode B), and by showing how the five-fence can be used for skip counting, the teacher enables students to discern the structure of numbers and the use of composed units (cf., Björklund et al., 2021b).

As shown by Murata (2015) and Karlsson & Wennergren (2014), using student errors as opportunities for clarification and reasoning can transform misconceptions into learning moments. In Teaching Episode B, the teacher used an incorrect answer to guide students toward a deeper understanding of numerical structure, maintaining focus on both content and participation, a balance that is pedagogically challenging (Machaba & Mangwiro, 2024; Ryve et al., 2013). As noted by Machaba and Mangwiro (2024), young children often express ideas in partial or non-verbal ways, requiring teachers to interpret and extend such input. The contrast between the two episodes illustrates how such interpretative work can constrain or expand learning opportunities.

Understanding how teachers respond to and elaborate on student input is important, as teachers’ real-time, context-sensitive ways of engaging with student input can influence what aspects of mathematical content become possible for students to discern (Ekdahl & Runesson, 2015; Maunula, 2018). This highlights the significance of not only what content is taught, but also how it is made visible through interaction. The findings of this study contribute additional empirical insights that enrich the broader literature on teacher–student interaction in early mathematics education. The data show that, although student input was frequently acknowledged, elaborated responses that supported reasoning and made mathematical content visible were only occasionally observed. This observation aligns with previous research (e.g., Emanuelsson & Sahlström, 2008; Hu et al., 2021). However, our findings add nuance by illustrating how differences in teacher responses, such as offering a non-example or contrasting representations, can open up learning opportunities affecting what becomes possible to learn.

While this study offers insights into how teachers respond to six-year-old students’ input in mathematics teaching, some limitations should be acknowledged. First, each classroom was observed only once, which limits the ability to capture variation in teaching practices over time. Second, the teachers were asked to design a lesson focusing on numbers and number relations leading to a variation in content focus and teaching design, which in turn opened up different possibilities for teacher–student interaction. Third, although all observations were coded using the adapted MPM framework, only two teaching episodes were selected for in-depth analysis using variation theory. These episodes were purposefully selected to illustrate qualitative differences in how teachers responded to student input. However, identifying such episodes was challenging, as we sought examples with similar mathematical content but contrasting teacher responses. As such, the in-depth analysis should be considered illustrative, highlighting possible variations in teacher responses.

Despite these limitations, this study offers important insights into how teacher responses can offer different learning opportunities in early mathematics education. The combination of MPM coding and variation theory provides both overview and depth, supporting the study’s aim to contribute to understanding the role of teacher responses and elaboration on student input in making mathematical content visible to young learners. Taken together, the findings underscore the importance of how teachers respond to and elaborate on student input, not only in terms of validating participation, but in offering students opportunity to discern aspects of the mathematical content and engage with key mathematical ideas. While many episodes in the data reflected a tendency to confirm correct answers without further elaboration, others demonstrated the potential of using student input as a resource for elaboration. These differences highlight the need for professional development that supports teachers in recognizing, responding to, and elaborating on young students’ mathematical thinking. Ultimately, it is through these real-time, context-sensitive interactions that teachers can transform student input into meaningful mathematical learning opportunities.

Conclusions

This study provides valuable insights into how teacher responses can offer different learning opportunities in early mathematics education, an area where research has so far mainly focused on older students. By combining MPM coding and variation theory, the analysis provides both overview and depth, supporting the study’s aim to contribute to understanding the role of teacher responses and elaboration on student input in making mathematical content visible to young learners. These findings underscore the importance of how teachers respond to and elaborate on student input, not only in terms of validating participation, but also in offering students opportunities to discern aspects of the mathematical content and engage with key mathematical ideas. While many episodes in the data reflected a tendency to confirm correct answers without further elaboration, others illustrated the potential of using student input as a resource for elaboration. These differences highlight the need for professional development that supports teachers in recognizing, responding to, and elaborating on young students’ mathematical thinking. Ultimately, it is through these real-time, context-sensitive interactions that teachers can transform student input into meaningful mathematical learning opportunities.