Introduction

We are intrigued with young children’s mathematical thinking. By understanding children’s insights and conceptions, we can challenge and develop their ideas. Often it is assumed that young children are too young to think multiplicatively. We wanted to test this assumption by offering 5–6-year-old children, experiences that would engage them physically and intellectually as they met ideas of equal groups for the first time at school. Our interest in how children think mathematically led us to contemplate whether children can consider a core concept in early multiplicative thinking — the simultaneous mental coordination of two composite units (quantifying groups of groups). For us to understand what was being imagined, we asked children to make models, to draw what they were thinking, and to talk to us about their experiences. We asked children in their normal mathematics classroom with their regular teacher to think about equal groups.

In earlier research, Anghileri (1989) claimed that the intuitive pathway for children to understand multiplication is through repeated addition. Indeed, as Bakos and Sinclair (2019) and Askew (2018) reported, repeated addition is commonly used for introducing and working with multiplication, and it becomes the dominant perception of multiplicative situations. Therefore, knowledge of counting sequences has been considered pre-requisite to quantifying equal groups. Sherin and Fuson (2005) described the ways in which some children use sequential counting in combination with unitary counting to solve multiplication problems. Some children count by ones and simultaneously keep track of the number of groups mentally to solve a multiplicative situation (Mulligan & Mitchelmore, 1997). In addition, Sullivan et al. (2001) showed that it seemed to be a relatively large cognitive step for children to move from using models with counting strategies, to abstract multiplication. The authors recommended approaches to the teaching of multiplication that required children of 5–8 years of age to imagine objects as well as model with objects.

We were keen to investigate what children notice and imagine when they meet multiplicative structures for the first time at school, so we established the Multiplication and Division Investigations (MULDI) project. The purpose of our study was to investigate young children’s thinking about equal groups when given thought-provoking tasks. Our challenge was to create learning contexts that would elicit children’s thinking so that we might understand what children recognise and visualise. The research question we set out to answer was:

What do young children’s responses to tasks reveal about their early multiplicative thinking?

Background literature

The background research literature for the study included the mathematical research findings concerning early multiplication and division, the features of mathematical tasks, the pedagogies of ambitious teaching, and understanding children’s thinking.

Early multiplication and division

Researchers of the development of young children’s concepts of multiplication and division have noted that these operations are conceptually demanding due to the levels of abstraction required and the complex semantic structures they involve (Anghileri, 1989; Clark & Kamii, 1996; Greer, 1992; Steffe, 1994). Being able to abstract or mentally view a collection of objects as a unit while simultaneously coordinating two composite units mentally is critical to children’s multiplicative thinking (Clark & Kamii, 1996; Steffe, 1994; Sullivan et al., 2001). For example, four times three requires: “the many-to-one correspondence between the three units of one and the one unit of three; and the composition of inclusion relations on more than one level” (Clark & Kamii, 1996, p. 42). The acquisition of this knowledge requires a shift in children’s thinking from additive reasoning because they need to understand a new set of number meanings that are related to multiplication and division (Clark & Kamii, 1996; Confrey, 1994; Nunes & Bryant, 1996). Siemon and Breed (2007) maintained that a shift from attending to how much is in each group (multiplicand) to how many groups (multiplier) is a critical step and leads to the recognition of the number of groups as a factor; this thinking supports generalising.

In summary, to understand multiplication and division a child needs to “coordinate a number of equal sized groups and recognise the overall pattern of composites (i.e., view a collection or group of individual items as one thing), such as ‘three sixes’” (Sullivan et al., 2001, p. 234). This is challenging for young children as it entails “using numbers to represent not the cardinal value of a single set but a mapping relation between two sets” (Sophian & Madrid, 2003, p. 1420). Sullivan et al. (2001) and Killion and Steffe (2002) suggested that the acquisition of an equal-grouping (composite) structure is at the core of multiplicative thinking. We agree that this thinking is central to multiplicative concepts.

Children’s strategy choice

Having examined what is required to understand multiplication and division concepts, we turn our attention to international studies in which children’s reasoning was examined. It has been reported that young children can solve multiplication and division problems by combining direct modelling with counting and grouping skills, and with thinking based on addition and subtraction (Anghileri, 1989; Becker, 1993; Bicknell et al., 2016; Carpenter et al., 1993; Kouba, 1989; Mulligan, 1992; Mulligan & Mitchelmore, 1997). The aforementioned studies were conducted prior to any formal instruction in multiplication and division and often in a one-to-one interview setting. Problems were either presented in a physical context (e.g., Anghileri, 1989) or on cards, and children could use materials to find solutions or construct a physical representation (e.g., Bicknell et al., 2016; Carpenter et al., 1993; Kouba, 1989; Mulligan & Mitchelmore, 1997). In some studies, it was reported that children used intuitive counting-based strategies to find a solution (Becker, 1993; Carpenter et al., 1993; Nunes & Bryant, 1996; Sophian & Madrid, 2003). These strategies included direct modelling (using physical objects, fingers, or drawings), counting by ones, one-to-many correspondence or many-to-one count strategy, and rhythmic counting. Both the many to one count strategy and rhythmic counting require children to mentally track the count across the groups, and use counting on. Beyond strategies that rely on a physical model, children employed: skip counting, which relies on their ability to abstract composite units, and doubling and halving, recalling known facts and derived facts (e.g., Anghileri, 1989; Carpenter et al., 1993; Mulligan & Mitchelmore, 1997; Sullivan et al., 2001).

In more recent studies, the accepted developmental trajectory of strategies has been challenged (Downton & Sullivan, 2017; Larsson et al., 2017; Sherin & Fuson, 2005). Askew (2018) found that when using appropriate models and contexts, children as young as 6 years old, could reason multiplicatively with functions “rather than only in scalar repeated addition terms” (p. 421). In addition, Sherin and Fuson (2005) reported that the emergence of number specific knowledge, such as 4, 12, and 32, is likely to influence strategy choice. In the present study, we created models and contexts in order to elicit children’s thinking about equal group structures underpinning multiplicative thinking.

Mathematical visualisation and drawings

As stated, the ability to form visual images of composite unit structures in multiplicative situations is fundamental to understanding multiplication and division. A feature of Becker’s (1993) study was that children were required to imagine or visualise the distribution of cookies. More recent studies conducted by Mulligan and colleagues (Mulligan et al., 2006, 2008) and Lu and Richardson (2018) also required young children to visualise a given structure (such as an array of dots), then to reconstruct it from memory or draw it. Mulligan et al. (2008) reported that drawings revealed the structure that children attend to, and highlighted the importance of imagining. These authors found that combining drawing with what the children have imagined was an alternative to requiring children to physically model situations. We agree that children can engage in multiplicative tasks that require them to visualise situations. For children as young as 5 years old, drawing and talking about their drawings can be alternative means of exploring multiplication (Cheeseman et al. 2020).

We have yet to find another study that has used story problems, without visual cues, or objects, to reveal how young children imagine equal groups to solve multiplication or division problems. In this study, we were focused on how children perceived, formed and represented, and enumerated equal groups. We used children’s drawings as a research tool to provide insights into their thinking.

Tasks

Clarke and Roche (2018) acknowledged the importance of tasks for mathematics learning. Researchers have also noted that the nature of student learning is determined by the task type, and the way it is used. Tasks underpin the interaction between teaching and learning (Christiansen & Walther, 1986). Researchers have suggested that, “instructional tasks and classroom discourse moderate the relationship between teaching and learning” (Hiebert & Wearne, 1997, p. 420). Questions that required high levels of mathematical thinking and reasoning were found to be related to the use of instructional tasks that engaged students in “doing mathematics or using procedures with connection to meaning” (Stein & Lane, 1996 p. 50). Non-routine tasks were found to provide optimal conditions for cognitive development. New knowledge is constructed when earlier knowledge is recognised and evaluated. Ames (1992) argued that students should see a meaningful reason for engaging in a task, that there needs to be enough but not too much challenge, and that variety is important. Fredricks et al. (2004), in a comprehensive review of studies on student engagement, argued that engagement is enhanced by tasks that are authentic, provide opportunities for students’ sense of ownership and personal meaning, foster collaboration, draw on diverse talent, and are fun.

The quality of tasks influences the nature of students’ interactions. Skinner and Pitzer (2012) noted that, “active participation, engagement, and effort are promoted by tasks that are hands-on, heads-on … intrinsically motivating, inherently interesting, and fun” (p. 29). In the present study, the features of tasks offered to children were informed by research, emphasising the importance of tasks that promoted thinking, encouraged mathematical discourse, highlighted mathematical reasoning, challenged and engaged children, and were fun.

While the features of tasks provoke opportunities for thinking, it is the ambitious teaching in the classroom that develops the potential of the tasks.

Ambitious teaching

The vision of instruction proposed in the NCTM Standards (2000) has been called an ambitious vision for high-quality mathematics instruction (Kazemi, 2008; Lampert et al., 2010). Jackson and Cobb (2010) emphasised the importance ambitious teaching to address issues of equity of access to mathematics. High-quality mathematics instruction was described by these authors in terms of student learning outcomes, the nature of tasks, the ways tasks were used, and the classroom norms established by teachers. Each of these aspects of ambitious teaching will be discussed in turn.

Students’ learning is the ultimate goal of ambitious teaching. Such learning has been described as the development of conceptual understanding of key mathematical ideas and procedural fluency in a range of domains (Kilpatrick et al., 2001). In addition to achievement goals, ambitious teaching enables students to develop mathematics argumentation, to learn to communicate their mathematical reasoning effectively by using multiple representations, and to make connections between different mathematical representations (Franke et al., 2007).

The nature of tasks, classroom norms, and mathematical discourse were also described in Jackson and Cobb’s (2010) ambitious teaching. Teachers using ambitious teaching pose cognitively demanding tasks that require students to explain their reasoning (Stein et al., 2000). In addition, tasks that “can be approached in more than one way” were identified as valuable (NCTM, 2000, p. 18). The use of such tasks included giving students time to work individually and/or in groups, and concluding the lesson with a whole class discussion in which students explain and justify their solutions. During the discussion, the teacher selects and sequences the solutions that are shared to ensure that the conversation focuses on key mathematical issues in order to advance the instructional agenda (Stein et al., 2008). Teachers press students to make connections between different solutions, and mediate the communication between students to help them understand the explanations that are offered (McClain, 2002).

Social norms and socio-mathematical norms established in classrooms with ambitious teaching guide and support student learning (Cobb et al., 2001). Potentially productive socio-mathematical norms require students to explain not only how they solved a task, but why they used particular methods, and how their solution differs from others’ solutions (Franke et al., 2007). The attributes of ambitious teaching elaborated by Jackson and Cobb (2010) apply to the present study.

Theoretical framework

Our research was based on a social constructivist learning theory, as we view “knowledge and competence as products of the individual’s conceptual organization of the individual’s experience” (von Glasersfeld, 1983, 2008, p. 48). We sought to discover how young children conceptualised multiplicative situations. We believe that young children construct their own knowledge in rich learning environments, and that problem-solving tasks create opportunities for children to think mathematically (Schoenfeld, 1992, 2014). In our view, it is essential to “observe children’s constructive processes firsthand … [allowing] a researcher the experiential base so crucial in formulating an explanation of those [constructive] processes” (Cobb & Steffe, 2011, p. 21). Like Schoenfeld (2014), we think “that research and practice can and should live in productive synergy, with each enhancing the other” (p. 404). We also acknowledge that, “real research is undertaken by those who teach children” (Anghileri, 2006, p. x). In line with these opinions, we designed a teaching experiment (Steffe & Thompson, 2000) to examine the complexities of early multiplicative thinking. To achieve our aim, we needed to find a way to understand what children think and how they organise concepts as they solve a multiplicative problem. We wanted to gain insights into children’s thinking as they engaged with tasks in their everyday classroom because, as Anghileri said, “the inner worlds of children repay close attention … [where] the process of learning is reviewed as the close and idiosyncratic involvement in events” (p. x). To elicit children’s thinking, and to learn about their ideas, we interacted with children and observed their actions, their words, and their drawings. We considered children’s drawings were important because through drawing children can communicate their mathematical thoughts (MacDonald & Lowrie, 2011).

A social constructivist theory influenced the research. We chose to study children learning in their mathematics classroom with their teacher facilitating mathematical conversations, expecting children to reason and justify their solutions. Our interpretation of the results was framed by careful consideration of children’s thinking as it was revealed by their actions, drawings, and explanations.

Understanding children’s thinking

The principal purpose of our research was to understand children’s early mathematical thinking. Steffe’s (1994) view of the importance of gaining insights into young children’s thinking echoed powerfully with us, as we believe that “to start from points accessible to the child and to establish these starting points seem indispensable to gain some insights into the child’s conceptual structures and methods” (Steffe, 1994, p. 5). In this study, we were concentrating on potential “starting points” of children’s early multiplicative thinking, particularly the process described variously as “constructing a composite set” (Steffe, 1994), “unitizing” (Lamon, 1994), “reunitizing,” and “reinitializing” (Confrey, 1994). In our teaching experiment, we closely observed children’s interactions to infer their thinking about multiplication as seeing “groups of groups.”

Method

A teaching experiment was conducted to explore and explain young children’s mathematical actions and thoughts about the equal group structures underpinning multiplicative thinking. Steffe and Thompson (2000) claimed the “primary purpose for using teaching experiment methodology is for researchers to experience, firsthand, students’ mathematical learning and reasoning” (p. 267). Teaching experiment methodology features exploratory teaching incorporating: interacting with students, knowing how to act, asking probing questions, and eliciting students’ reasoning in a responsive and intuitive way. A teaching experiment involves a sequence of teaching events with four components: a teacher (researcher), students, an observer, and a way of recording what occurs during the teaching event (Steffe and Thompson). The role of the observer is to focus on students’ reasoning and actions. How students develop conceptual understanding is noted, and the next phase of teaching is considered. Records of events are used to prepare later events and to conduct a retrospective conceptual analysis of the teaching experiment. A teaching experiment is cyclical and provides a researcher with a better model of students’ mathematics understanding over the sequence of events.

The MULDI project included each of the defined components of a teaching experiment, as it was a sequence of mathematics lessons over of five consecutive days in one school with one class of students in their first year of formal schooling. The classroom teacher, Sarah (a co-researcher, and fourth author), facilitated the learning; three university researchers observed and video-recorded each lesson.

Consent for collecting and publishing video recordings was gained from parents, the teacher, and school principal, and the reason researchers were in their classroom was explained plainly to the children1.

Participants

Participants in the study were 21 children (13 girls and 8 boys) from a school in a large rural city of Victoria, Australia. The children’s mean age was 5 years 6 months which was a little younger that the mean age of 5 years 9 months for Australian children in the first year of school (Thomson et al., 2005). The school demographic was largely middle-income families. The teacher (Sarah) and her class provided a convenience sample for investigating our research questions, as the children had not had any introduction to early multiplicative thinking at school. Sarah was an experienced primary school teacher (21 years) with a doctorate in Mathematics Education.

Research context and procedure

Sarah expressed an interest in investigating the conjectures raised by the project and participated in the theoretical framing of the study. Independent of Sarah, the university research team created and collected potential tasks for her classroom use. Immediately prior to the commencement of the teaching experiment, twenty lesson outlines were emailed to Sarah for her evaluation and consideration with her children. The whole research team met to discuss our preferences and to decide which five lessons to use and the order of the lessons. We gave priority to Sarah's choices because she had expert knowledge of the children.

Teaching experiment overview: the lessons

The teaching experiment included various thought-provoking tasks including a dancing game, tasks with cubes, painting, subitising with cards, and problem solving. An overview of the sequence of five lessons is shown in Table 1 to enable the reader to see the broad picture of the research. In the “Findings” section of this paper, each lesson is described more fully and paired with illustrations of the children’s work and thinking.

Table 1 Overview of the sequence of lessons, the context and mathematical focus
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It should be noted that the mathematics content of the five lessons (see Table 1) was beyond the intended curriculum for Foundation level of the Australian Curriculum: Mathematics (ACARA, 2013) where there is no mention of learning outcomes related to multiplication. There is one outcome statement that mentions “sharing.”

In addition to analysing the mathematical content and positioning it within the curriculum, the research team evaluated and summarised pedagogical features of each lesson (see Table 2). Features of the lessons reflected the values we hold as mathematics educators: tasks that elicit children’s thinking, target our intended learning outcomes, use meaningful and engaging contexts that invite children to actively participate, promote mathematical communication, involve problem solving by young children, offer student choice, and use mathematical models that serve children. Our tasks addressed many of these attributes.

Table 2 Features of MULDI Lesson
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The mathematical content was considered likely to present conceptual challenges for some children aged 5–6 years (in their first year of school). In this regard, the lessons represent ambitious exploratory teaching (Jackson & Cobb, 2010).

Pedagogical approaches

The researchers held similar constructivist theories of learning and pedagogical beliefs and values. These theories have classroom implications in terms of our expectations of children as active builders of knowledge and for teachers as initiators of experiences, motivators of participation, and instigators of children’s thinking (Jackson & Cobb, 2010). In pedagogical terms, we value the features of ambitious teaching, in particular encouraging children to communicate their mathematical reasoning effectively using multiple representations and explain how their solution differs from those of others (Franke et al., 2007).

The pedagogical approaches taken by Sarah match those described as ambitious teaching by Jackson and Cobb (2010). Sarah was familiar with the potential of challenging tasks (Downton & Sullivan, 2017), and differentiated instruction with enabling and extending prompts in her classroom (Sullivan et al., 2006). She structured her problem solving lessons in phases of Launch, Explore, and Summarise in ways similar to those described by Lappan and Phillips (2009). She also enacted her conviction that solution strategies should not be demonstrated to children before they attempt a task. Sarah knew the research of Laine et al. (2018) in which it was shown that modelling solutions in the Launch phase of a lesson did not achieve improved results. Sarah used hermeneutic questions, and often required higher order thinking of children (Davis, 1997). She listened to students empathetically, and asked the children to explain their thinking (Davies & Walker, 2007). Her instructional practices were productive pedagogies that focused on student learning. These actions also aligned with the exploratory teaching described by Steffe and Thompson (2000).

While the children worked on the task in the main Explore phase of each lesson, Sarah and the researchers observed, listened, and asked questions to gain insights into the children’s thinking. Examples of probing questions included: “Can you tell me about what you have been doing?” “What have you drawn?” “Why do think…? The children’s responses were recorded as data.

Data collection

To achieve our aim of investigating young children’s thinking about equal groups, we decided the main source of data would be the children’s actions, their drawings, and their talk with us, during the lesson. To capture classroom events, we collected six different sets of data as shown in Fig. 1.

Fig. 1
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Data sets collected for each of five lessons

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We had two video cameras mounted on tripods in fixed positions: one camera focused on the classroom meeting space and another camera focused on the children’s work tables. These fixed cameras recorded events from a distance and provided a general picture of the classroom. Additionally, three tablet cameras operated by the observer-researchers recorded children working or in conversation with an adult. The researcher-observers maintained coverage of the classroom by moving through the classroom at a distance from each other. As the children drew their solutions, we video-recorded their actions and their explanations, took photographs of their work in progress, and collected their finished work. We also collected data after the lesson; the university researchers’ audio-recorded verbal reports, and the teacher submitted written reflections via email.

Data analysis

There were three phases of data analysis. The first phase happened shortly after the lesson when audio-reports by the university researchers used field notes, photos, and video excerpts to describe in detail what each person had observed. We shared our experiences of the classroom and developed a multi-perspective understanding of the day’s events. The analysis of the children’s mathematical thinking began with our experiences. We observed children’s actions, listened to their explanations, and noticed the way they represented their solutions. In this way, we built a complex picture of the events in the lesson, began to theorise our experiences, made summary notes of our initial thoughts, and organised and annotated the data sets.

In the second phase of data analysis, we tested our observations by systematically interrogating the recorded data. Reviewing the data in this way enabled us to validate or contradict our observations, re-examine our initial conjectures, and to consider whether the findings were answering our research question. Daily reflection enabled us to triangulate the sources of data (observations, videos, children’s drawings, and teacher reflections) to increase our confidence in the validity of the findings.

The third phase of analysis involved a retrospective analysis of all the data collected over the teaching experiment. We “stepped back” to take an overview of the events in the classroom. In particular, we noted events of interest. These episodes or exchanges were occasions where children’s multiplicative thinking was evident. The events might have involved highlights or something interesting a child had said or done. We found occasions when children’s thinking had features in common. We also noted contradictory events to discuss as a team. We identified the place(s) in the different data sets that served to triangulate the data. We compared and contrasted evidence we had gathered to consider what had been learned, the extent to which we had addressed our research question, and any questions the teaching experiment had raised.

Events of interest have formed the evidence we have presented to illustrate and exemplify the findings in this paper (see for example, the photo and transcripts of exchanges in Lesson 1 on page 10). In this way, we selected events that focused on ideas spontaneously expressed by children that illustrate their thinking during lessons.

Findings

We present findings using excerpts of the data and descriptions of events, together with some quantitative analyses to illustrate the children’s thinking during the teaching experiment. Each lesson’s findings are briefly set in context, and data from the various sources have been selected to demonstrate the range of the children’s mathematical thinking that were captured and identified. Findings from the lessons are presented in the order in which the lessons were taught.

Lesson one Part 1: Go-Go Groups

  • Problem: Can you make equal groups?

  • Rationale: To physically model equal groups of given numbers in the range 2–10.

  • Sources of data: Children’s drawings, transcripts to illustrate classroom interactions.

Sarah introduced the activity by saying she would play the guitar, and when the music stopped a “Go-Go Group” number would be called out, and everyone would get into a group of that number, for example, “Go-Go Groups of two.” Immediately, Finn (pseudonyms are used throughout) raised his hand and said, “That would work out but what if it didn’t?” There were 20 children in class that day. Finn knew 20 could be grouped in 2 s as can be seen by his remark “that would work out” but he wanted to know what would they do if the class was not divisible by the number called out. Sarah said, “People not in groups will come and stand with me. They can call the next number.” Finn was satisfied. He had envisaged the eventuality of not being able to form equal groups and some people being left out. His question showed that he intuitively realised division could produce remainders.

As the game began, other children displayed their thinking. They made groups of three, five, four, and two, all the time counting and checking as the excitement rose (see Fig. 2). It became clear that some children could imagine numbers that were divisible and those that would produce remainders. Freddie was waiting to call the next Go-Go Groups number. He said that he wanted “one that does not go.”

Freddie: I think seven. Seven will not go into groups will it Sarah?

Sarah: Are you thinking about seven? You could try it to see what happens.

Fig. 2
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Children dancing and making groups

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The children took some time to form groups of seven and to check them. One group had six. Freddie smiled to himself and said quietly, “I thought seven would not work.” Although we cannot be positive, it seems that Freddie was trying to figure out multiples of seven mentally. While mention is made in the Australian Curriculum: Mathematics (ACARA, 2013) about counting by 10 s, 5 s, and 2 s as a learning outcome for 6–7 year-olds in Foundation year, nothing is written about multiples of seven until Year 2 when the outcomes include: “Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031). Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)” (n.p.).

The last Go-Go Group of the day was called by Ellie as 10 s. She could not explain why she chose groups of ten. However, Mia could explain why Ellie’s “Go-Go Groups of 10” was a good choice.

Mia: Because ten and ten is twenty. That would make 10 groups.

Sarah: Ten groups. [a long pause, apparently thinking] How many groups?

Mia: Two groups made out of ten people.

Sarah: Two groups made out of ten people. That is a good way to say it. Mia.

[Nancy’s hand was raised.]

Sarah: Nancy?

Nancy: When you count in ten it goes ten, twenty. That’s how you know it will make two tens and that will make twenty.

Sarah: Wonderful. Finn have you got something to add?

Finn: One half is ten and another ten to get to 20.

Sarah: Did you say half? [pause] Half! [pause] That was an interesting way to think of it.

As Lisa’s drawing of Go-Go Groups shows (Fig. 3), she understood their explanations.

Fig. 3
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Lisa’s drawing of the Go-Go Groups game

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Summary of findings

These children were:

  • thinking about the numerical relationships between 10 and 20;

  • adding two equal groups of ten to form the total of twenty;

  • considering counting by 10 s, and thinking of part-whole relationships;

  • calculating mentally; and

  • showing mathematical reasoning and number sense.

This activity was so much fun that children kept asking to play Go-Go Groups in subsequent lessons. On Day 5 at the end of the lesson, the excitement was palpable when Sarah fetched her guitar. There were 17 people at school that day. They played with groups of three, and Rachel was asked to count the groups of three point-counting to get “five.” SarahX commented, “Good job, five groups of three and two people were left over.” Next Rachel had her turn and she smiled as she chose her number.

Rachel: Go-Go Groups of zero.

[… The children scattered.]

Sarah: What do you think will happen?

Rachel: Everyone will hide.

Sarah: I don’t expect to see anyone if we have groups of zero.

The children giggled as they found places in the classroom to hide. They enjoyed Rachel’s joke. It was time to draw the session to an end when the following exchange took place.

Sarah: One last turn.

Nancy: No one can miss out.

Sarah: Nancy says that no one is allowed to miss out. Remind me, how many beautiful children are here today?

Chorus: Seventeen.

Sarah: What number could I choose so I know that no one is going to miss out?

[Suggestions were made of 2, 3 and 6.]

Sarah: Lots of ideas. Let’s try twos then.

…

Twos didn’t work did it? One person missed out.

Sarah: What number could I choose so I know that no one is going to miss out?

Freddie: Seventeen!

[All of the children clustered together - most amused.]

Sarah: Is there anything else that would definitely work?

Freddie: Go-Go Groups of one would definitely work.

Sarah: Okay let’s see that. … Everyone standing on their own!

Summary of findings

The second iteration of the activity revealed that individual children could:

  • focus on the composite groups;

  • transfer from counting people in a group to counting groups;

  • interpret “groups of zero”; and

  • recognise that 17 could be divided equally in two ways.

Lesson one Part 2: gathering groups’ game

  • Problem: Can you gather and show equal groups of cubes?

  • Rationale: To use cubes to model groups. To sort by colour and to display cubes by group size.

  • Sources of data: Video recordings and observations

The Gathering Groups activity was a cooperative table game involving sorting cubes and displaying the colour groups ready to play (see Fig. 4). A standard 6-sided dot dice was rolled to determine the size of the group of cubes to be gathered in the centre of the table. For example, the dice rolled 6, and all players placed their groups of 6 in the centre of the table (see Fig. 5). The total number of groups was recorded by a researcher on a sheet of poster paper (see Fig. 6).

Fig. 4
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Preparing colour groups

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Fig. 5
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Gathering groups according to the roll of the dice

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Fig. 6
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Reading the records of the groups they had gathered

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Finn commented as he sorted his cubes ready to play, “I’ve got two threes.” It was fascinating to watch Finn “point count” by ones on each occasion as he played this game, whereas in other situations he was working abstractly and visualising numbers. It raised a question about the types of manipulatives and models used with early multiplication, and which seem to prompt counting strategies for some children.

The transcript of exchanges towards the end of the game showed children saying, “three sixes” and “five threes” to describe the collections they formed and placed in the centre of the table. Ellie thought it was interesting that there were five groups of five and dictated what to write on the poster as a record. The poster was shown to the children and they were asked, “What numbers on the dice do we still need to roll?”.

Summary of findings

It was apparent that children:

  • understood the symbols and the words used to describe their results;

  • knew that the dice roll determined the size of the groups;

  • quantified of the number of groups of a given number; and

  • understood the game was quantifying equal groups.

Lesson two: handprint posters

  • Problem: Can you make posters of handprints with paint?

  • Rationale: Using composite units of five and recognising how many “high-fives”.

  • Sources of data: Observers’ field notes and the analysis of video data.

The lesson began with Sarah modelling how she “collected high-fives.” As a child tapped the palm of her hand with theirs, she made a little exclamation of delight and said, “one high-five, two high-fives …”; she invited the children to collect high-fives and to remember how many they had collected (see Fig. 7).

Fig. 7
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Children collecting high-fives

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The children were asked to state how many high-fives they had collected, and Sarah repeatedly modelled saying the unit, for example, “seven high-fives.” The children followed her model and, as they did so, she occasionally said, “four fives – like me!” dropping the “high-five” and calling them “fives.”

Next, Sarah explained that they were going to make handprints in paint to show high-fives on paper (see Fig. 8).

Fig. 8
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Ellie’s handprints of seven high-fives

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The following day the lesson began with the children being asked to add to their posters by “writing some numbers on them to help us to see what you have done.” In a sharing circle, Sarah challenged children to say what numbers they had used and why. Some children had labelled the high-fives, 1, 2, 3, etc. (composite units) and counted them that way (see Fig. 9). Others had labelled the fingers in groups of five, 5, 5, 5, etc. (see Fig. 10), and some had simply written the total number of handprints/high-fives on their posters. The children were assigning numbers to the size of the group and the cardinal number of the groups.

Fig. 9
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Harry numbered each handprint

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Fig. 10
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Rachel labelled five fingers on each handprint

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Sarah listened to children’s explanations and asked, “Whose poster is this like? Why?” Through these questions children were being asked to consider mathematical similarities and differences in the purpose of the numbering.

The transcript revealed Lisa’s description of what was on her poster. She had labelled each handprint with the numerals 1 to 9 to enumerate composite units. She was coordinating two levels of units to count the number of times a composite unit had been recorded.

SarahX: So, your poster is like whose?

Lisa: Like Harry’s.

Sarah: It is like Harry’s. So, this one here [Sarah is pointing at one handprint] is what? [pause … silence] Is it one high-five?

Lisa: Yes. It goes to nine fives on the poster.

The last activity with the posters involved ordering posters from those with the least handprints to those with the most (see Fig. 11). Here, Sarah was focusing the children’s attention on enumerating the composite unit.

Fig. 11
figure 11

Posters in order from least handprints to most handprints

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It was apparent in this lesson that many children could think about fingers and hands as a natural composite unit for fingers. This idea of being able to “see” the unit and the composite unit simultaneously is known to be a conceptual step for children (Ulrich, 2015). In addition to the composite unit thinking, the way Sarah made links to new vocabulary was also a feature this lesson. The everyday “high-fives” language became the mathematical “fives” in a natural and logical way. Children were talking spontaneously about “four fives” for example, by the end of the activities.

Summary of findings

Children could:

  • conceptualise a unit and a composite unit simultaneously; and

  • use mathematical language about multiplication spontaneously.

Lesson three: Seeing dots and collecting sets of dotty cards

  • Problem: What can you find out about your dotty cards?

  • Rationale: Subitising dots to five, matching different arrangements of the same quantity, collecting sets of dot-cards, and counting the cards in each set. Simultaneously, noticing dots as one unit and cards as another unit.

  • Sources of data: Video evidence and observer’s records.

The beginning of this lesson connected to the previous day’s work of making handprint posters. Each child was given a bag of prepared cards to investigate (see Fig. 12).

Fig. 12
figure 12

Dotty cards

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Two sets of cards were available to Sarah. One set had the more common dice dot arrangements, and the other had non-conventional arrangements of one to five dots. There were blank cards for children to make their own “dotty cards.” Sarah chose to use all of the cards. The children were invited to investigate the packet of cards and to sort them. Then the children were taught a solitaire game that required turning over and matching two cards with an equivalent number of dots. Pairs of cards were set aside to be counted and described later.

We observed variation in children’s ability to subitise in ways that were similar to those reported in the literature (Clements & Sarama, 2009; MacDonald & Wilkins, 2019). It is known to be easier to suddenly recognise the numerosity of particular arrangements of dots (Clements & Sarama, 2009). Typically, small collections of fewer than five are easier to see instantly. The arrangement of collections also influences children’s subitising. In general, the difficulty from simplest to most complex is lines of objects (small numbers), rectangular shapes, then dice arrangements, and lastly different arrangements and larger numbers. Clements and Sarama also described children’s thinking as developing from perceptual subitising where children “just see” to conceptual subitising where they “just know.” At the age of 5 years, these authors characterise children’s thinking to be transitioning to conceptual subitising. Further, they claimed that subitising is “one of the main abilities young children should develop” (Clements & Sarama, 2009, p. 9). However, as MacDonald and Wilkins stated, “it is not yet clear the role subitising has in children’s early construction of units” (2019, p. 3).

The cards were seen by many children as units of dots; for example, Mia, when she played the game, instantly recognised five in its dice arrangement but paused to count silently (without pointing) the non-standard arrangements of five dots (see Fig. 13).

Fig. 13
figure 13

Mia pausing to count five in a non-standard arrangement

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We could see, by the longer pauses, that Mia did not instantly perceive non-standard arrangements of five dots.

Having focused on sorting cards with the same number of dots on them in the main part of the lesson, in her summary, Sarah drew the children’s attention to the different ways they had drawn their own cards of five. As each child described what they had drawn, Sarah recorded the various arrangements (see Fig. 14) and asked others what they “saw.” She then asked whether there were cards that were alike. The children described the similarities they could see. For example, one child said that the first and the last card were alike because she saw “two and three” and “three and two.” Sarah was encouraging conceptual subitising in the closing stage of the lesson.

Fig. 14
figure 14

Sarah recording of different five-dot cards

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Summary of findings

Children could:

  • notice dots as one unit and cards as another unit simultaneously;

  • match and sort cards by subitising or counting non-standard arrangements of dots;

  • create sets of equal dots;

  • enumerate the sets of sorted cards; and

  • consider the number of different ways that cards of five dots could be drawn.

Lesson four: twelve little ducks

  • Problem: Can you organise 12 little ducks into equal groups?

  • Rationale: Multiplicative thinking and visualising 12 in equal groups.

    Sources of data: Quantitative analysis of the students’ work samples.

To introduce the task to her children, Sarah told a story about a mother duck with twelve little ducklings:

In order not to lose any of her ducklings, the mother duck put them into some groups that were the same. She put them into equal groups, because it was easy for her to see that she still had her 12 baby ducks. Can you make a picture in your head of those 12 little ducklings? The mother duck put them into groups with the same number of ducks in each group. I wonder what groups she put them into. I would like you to draw a picture of what is in your head. We are interested in your thinking about equal groups, and if you draw a picture we can see what you are thinking (video transcript).

Sarah described why she chose not to show a picture of ducks or to model the problem with materials. She was keen to learn what her children could imagine without objects in a context the children would understand and find engaging. As her reflection showed, Sarah was also very conscious of the challenge of the mathematical vocabulary in the task:

These children have not heard the term “equal groups” from me at school at all until today. I did say “the same number in each group,” but I did not go into great detail about what I meant by equal groups (diary reflection).

These pedagogical decisions deliberately created a “challenging task” (Sullivan et al. 2014) for these 5 to 6 year-olds. Objects were not provided initially, but five children were offered blocks to support their thinking when it was apparent they were struggling to find a solution. Subsequently, two of these five children solved the problem.

We analysed work samples alongside observations, recordings of children’s explanations, and video data to quantify children’s multiplicative reasoning. Three distinct categories of demonstrated multiplicative thinking could be described. The most proficient thinking was characterised as Evident where 44% (n = 8) of the children could simultaneously quantify objects in groups and enumerate groups as new units. We described a further 39% (n = 7) as demonstrating Partial multiplicative thinking, where children could quantify the objects in groups but not enumerate the groups as new units. The final category we termed Emergent where 17% (n = 3) of the children were unable to solve the problem by producing equal groups. For further detailed description of the thinking revealed by children and the scale developed by the authors (see Cheeseman et al. 2020). Taken overall, the results show that 83% of the children had acquired the equal-grouping (composite) structure that Killion and Steffe (2002) argued is at the core of multiplicative thinking.

There were individual children who noticed numerical relationships as they solved the problem. For example, Sarah noted in her reflection that Finn drew three groups of four then four groups of three. As he was telling her what he had drawn he noticed a pattern in the numbers and exclaimed, “Three groups of four—four groups of three. Hey! That’s cool!” This child was noticing a pattern in multiplicative structures; that is, that multiplication is commutative. It may be some time before the initial noticing becomes formal knowledge, but he discovered something exciting in grouping 12 ducks.

We found that some children pre-planned the organisation of their groups on the page, and were observed circling that group before drawing the next equal group. This was reminiscent of Sfard’s (1991) claim that mental pictures make abstract ideas more tangible, as they can be manipulated, almost like real objects. Other children drew 12 ducks first, and circled groups after all ducks were drawn and, we conjectured, the drawn ducks were treated as objects with which to form equal groups (Sherin & Fuson, 2005). Furthermore, Bobis (2008) found that young children are only able to abstract a composite unit when they have constructed meaning in their own minds.

Summary of findings

The results show that children:

  • used equal-grouping (composite) structure;noticed pattern in multiplicative structures, that is, that multiplication is commutative; and

  • manipulated abstract mental objects or manipulated their drawings to find a solution.

Lesson five: things in boxes

  • Problem: Can you work out the number of turtle legs on four toy turtles inside two small boxes?

  • Rationale: Have children visualise hidden equal groups as a simple rate problem, that is, four legs on each turtle.

  • Sources of data: Video data and work samples.

Sarah showed the children a large soft toy turtle (see Fig. 15). She passed the turtle around the group of children and asked them to look closely at it and to notice how many legs the turtle had. This was to establish the knowledge needed to solve the problem. For a detailed account of this lesson and the implications for teaching, see Roche et al. (2021).

Fig. 15
figure 15

Setting the context of the Things in Boxes problem

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Sarah introduced the task saying:

I have this special green turtle-coloured box. And this box is where I am packing away my small toy turtles. This turtle is big (see Fig. 15), but the ones I am packing away in the box are small. Inside this green box, I have two smaller boxes [gestured with her hand on the lid to demonstrate where they fit in the green box] and in each of those smaller boxes are two turtles [held up two fingers]. I want to know how many turtle legs, toy turtle legs, are in this box. I want you to draw a picture of what you are thinking (video transcript).

Fifteen children (approximately 80%) had drawn a solution to the main problem that included four turtles with 4 legs each and all of these children had recorded the number 16 on their drawing. The ways they had found the correct solution varied. Some children drew turtles and counted each leg. However, Lisa had written her answer of “sixteen” in mirrored digits (61) prior to drawing the turtles (see Fig. 16). We conjecture that Lisa had solved the problem mentally. Cara used a different approach; she used her fingers to show two groups of eight (see Fig. 17).

Fig. 16
figure 16

Lisa’s drawing of 61(sic) was a representation of her abstract thinking

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Fig. 17
figure 17

Cara’s solution showing two groups of eight

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To imagine turtles in boxes, children were expected to visualise and to draw their mathematical thinking. Eighty percent of children found a correct solution to the two turtles with four legs in two boxes problem. In addition, half of the children could draw the extension problem of four turtles in each of two boxes accurately, although only about one third of the class could solve 4 × 4 × 2 correctly. We conjecture that perhaps the solution of 32 was beyond the number knowledge range of many children. The children’s responses to the original problem showed that as a large majority of children could think abstractly about multiplicative situations involving a simple rate problem (children were picturing four legs on each turtle and two turtles in each box). While this finding echoes that of Anghileri (1989), the complexity of the problem is beyond those used in her work, as Things in Boxes was a two-step problem that required visualisation and abstraction of multiplicative thinking about rate. We have found no other literature in which such findings with young children are reported.

Summary of findings

The findings show that children could:

  • abstract a multiplicative situation involving a simple rate problem; and

  • solve a two-step problem that required mathematical visualisation.

Discussion

We established the MULDI project conjecturing that many children recognise equal group structures when contexts encourage them to. We investigated what children noticed and imagined when they met multiplicative structures for the first time at school. The research question we set out to answer was: What do young children’s responses to tasks reveal about their early multiplicative thinking?

Taken as a whole, the early multiplicative idea of equal groups can be communicated by children through drawing and talking. Given contexts that make sense to them, children can understand foundational ideas of multiplication. Our findings indicate that 5–6-year-old children can imagine equal group structures and, in doing so, recognise composite units (Sullivan et al. 2001). Based on their experiences, children seemed to have intuitive understandings of equal group structures because they came to the problems we posed without any prior formal instruction about equal groups. Further, some children could enumerate the composite units as was illustrated in Lesson 4 (Twelve Little Ducks) when almost half of the class demonstrated this thinking. To have a complete understanding of early multiplication, Clark and Kamii (1996) argued that children need “to be able to think about units of one and about units of more than one” (p. 43). We know that some children in the present study exhibited this core concept of multiplicative thinking — the mental coordination of two composite units. We assert this thinking is foundational to later sophisticated multiplicative reasoning.

Children communicated their thinking about equal group situations through their drawings, and elaborated their meaning with verbal descriptions and gestures. Our findings were reminiscent of Askew (2018) who found that “using appropriate models and contexts, children as young as 6 years old…can begin to reason in multiplicative and simple functional terms…” (p. 421). We argue that the present findings push the claim to include even younger children as our participants were between 5 and 6 years of age. We contend that with meaningful models and context, our participants could reason multiplicatively. This finding adds to the extant literature. In our searches, we have found no studies in which similar results with 5–6-year-old children have been reported.

Children’s drawings represented abstract thinking and call into question the accepted view of the way early multiplication typically develops. The standard orthodoxy is that children progress from a reliance on direct modelling to solve tasks, to partial modelling, then to developing multiplicative thinking where they are operating on problems abstractly (e.g., Anghileri, 1989). We argue that it may be productive to require young children to abstract problems earlier. We think that requiring visualisation, together with drawings, is an alternative approach to direct modelling. Manipulatives can be used as a support to transition from physical situations to abstract thinking as Sullivan and his colleagues (2001) noted. However, we think that further detailed investigation is needed of the types of manipulatives, and physical experiences, that help children conceptualise early multiplication.

Implications for the classroom

In this study, we have shown that there is a need to heed the advice of Clarke et al. (2002), who identified a barrier to abstracting multiplicative problems, and advised that by focusing on appropriately chosen activities, teachers could assist more students towards abstraction. The key to our experience was the appropriate activities and the playful mathematical approaches that were used to provoke children’s thinking. The findings of this study have implications for teachers, as they reveal the importance of:

  • creating learning provocations that elicit children’s early ideas of multiplication;

  • encouraging visualisation and abstraction by children;

  • using thought-provoking meaningful tasks that reveal children’s thinking; and

  • observing and listening to children’s explanations of their thinking — possibly using insights provided by their drawings.

In conclusion

We acknowledge the limitations of this small study: it was conducted in one classroom of one school, with only 21 children over a short period of time. We make no claims of generalisability of the findings, nor is the study based on a representative sample of 5–6 year-olds. However, we do think that as an initial investigation of children’s thinking before they had received any formal school instruction, it showed some interesting results. We found that children’s responses to tasks revealed that many young children could think about, visualise, and draw, multiplicative situations involving equal groups.

We encourage others to continue to investigate young children’s mathematical thinking. Like all research, the study has raised questions. In future, researchers might investigate other contexts that elicit young children’s thinking about key ideas underpinning multiplicative reasoning. During this research, we queried whether there are manipulatives used with early multiplication that prompt counting strategies for some children. We wonder whether these counting situations support or hamper transitions in thinking to more sophisticated reasoning. Another question raised is the role subitising has in children’s early construction of units. These questions warrant further investigation. Children’s drawn representations of their mathematical reasoning continues to be an area of interest for researchers, and the intuitive understanding that young children develop about aspects of multiplication are worthy of detailed examination in the future.

The findings from this study make an original contribution to the literature and offer reasons to re-examine some prevailing views of ways to initiate young children’s multiplicative thinking. Insights were gained into 5–6 year-olds’ ability to conceptualise core aspects of multiplicative thinking, in particular, the mental coordination required to quantify groups of groups. These ideas become essential to the development of multiplicative reasoning. The results emphasise the value of: encouraging mathematical visualisation from an early age; using open, thought-provoking meaningful tasks to reveal children’s thinking, and promoting drawing as a form of mathematical communication.