NZ J Educ Stud DOI 10.1007/s40841-016-0073-9 ARTICLE

Investigating Children’s Thinking About Suspended Balances Jill Cheeseman1 • Andrea McDonough2 Dianne Golemac1

•

Received: 6 January 2016 / Accepted: 12 December 2016  New Zealand Association for Research in Education 2016

Abstract With limited prior research on young children’s learning of the measurement of mass, the study reported in this paper provides insights that can usefully inform teaching. Close examination of the actions and conversations of 12 children of 5–7 years of age, as they experimented for the first time with suspended balance scales, led to the identification of themes presented in this paper through focused discussion of the explorations of five of the children. Stimulated by exploration and investigation of the tool, children were creative in figuring out how the tool worked, reasoned mathematically, and gave attention to some ‘‘big’’ ideas of mathematics. They were also able to transfer knowledge, express notions of equivalence, mathematise and generalise from their experiences. The study emphasises the importance of eliciting children’s mathematical reasoning, the value of attending to what children notice, and the need for careful and specific use of comparative terms. Keywords Mathematical reasoning
Equivalence
Investigative play

& Jill Cheeseman [email protected] Andrea McDonough [email protected] Dianne Golemac [email protected] 1

Monash University, Melbourne, Australia

2

Australian Catholic University, Melbourne, Australia

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Introduction Children’s thinking is important because new learning is built on existing thinking. In education we often create contexts for learning by offering children tools to use. Knowing how children understand the tools they are given helps us to maximise their opportunities to learn. The use of equipment, or manipulatives, is an integral part of mathematics in the early years of school. These materials are intended to physically model situations, to demonstrate mathematical concepts, and to move children’s thinking from the physical to the abstract. Often teachers have clear pedagogical purposes for the tools they select. However, what is not so clear is what children ‘‘see’’ in the tools they are provided. Measurement, including the measurement of mass, is internationally recognised as a key element of the mathematics curriculum (e.g., Australian Curriculum and Reporting Authority, ACARA 2014; Common Core Standards Initiative 2014; Department for Education 2014; Ministry of Education Singapore 2014; New Zealand Ministry of Education 2014). However, there remains insufficient research in this area (Sarama et al. 2011; Smith et al. 2011). With measurement being a key, but under-researched area of the learning of mathematics, we focus on children’s thinking about weighing to understand how young children investigate balance scales. Our research question was: How do children experiment with suspended balances and what do they come to understand as a result of their testing? No prior research was found pertaining to these questions. The type of scale is not common or in everyday Australian usage. We were interested to know whether the weighing tool is understood by children and whether they can transfer their understanding of comparison of mass, by holding an object in each hand, to this new context. This paper reports a small study of 12 5–7-year-old children’s thinking.

Theoretical Background The theoretical framework for the research is a social constructivist perspective that was summarised by Ernest (1994) as recognising that knowing is active, ‘‘individual and personal, and that it is based on previously constructed knowledge’’ (p. 2). Ernest believed that teachers need to be sensitive to learners’ previous constructions. Consistent with this view, Nescher commented that conversations between teacher and child: … serve many purposes. From the constructivist point of view, it is the major means by which the teacher has the opportunity to learn about the student’s thinking and have a real dialogue (Sfard, Nescher, Streeflend, Cobb, & Mason, 1998, p. 43). In accordance with the social constructivist emphasis on understanding and building on students’ thinking, we believe it is of value for teachers and researchers to listen and to come to know students’ current and developing understandings, so as

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to be able to appropriately respond and challenge children’s mathematical thinking. We also believe that through the provision of rich learning opportunities in mass measurement, children can develop understandings of measurement tools and concepts (Cheeseman et al. 2014). The study reported here, which was developed with a social constructivist orientation, explored young children’s developing understandings of the measurement of mass in a situation where they explored a tool that is not commonly used in Australian schools or everyday contexts. Eliciting and Valuing Children’s Thinking Listening and responding to children is this research involved noticing ‘‘when powerful new connections open up for individuals or groups’’ (Hipkins 2009, p. 15). As Lansdown (2004) advised we listened ‘‘to learn to hear and see what children are saying and doing without subjecting it to a filtering process that diminishes their contribution’’ (p. 5). Together with other researchers we consider children as experts in their lives and we find ways to allow children to share their thoughts to achieve a better understanding of children’s perspectives and competencies (Clark 2007; Formosinho and Araujo 2006; McDonough and Sullivan 2014; Schiller and Einarsdottir 2009). One of the purposes of this research was to understand children’s mathematical perspectives and competencies through conversational exchanges. Eliciting children’s thinking to encourage mathematical communication has been discussed extensively in the literature (Cheeseman 2009). The focus here is on conversation which was described by Thornbury and Slade (2006) as: … the informal, interactive talk between two or more people, which happens in real time, is spontaneous, has a largely interpersonal function, and in which participants share symmetrical rights (p. 25). Sfard et al. (1998) focused on the importance of dialogical interactions centred on mathematical ideas which promote the personal active construction of mathematical meaning through communicative processes. Nescher described the interactionist argument—the idea of learning through conversation as a natural by-product of the conception of learning as an initiation into a ‘‘community of practice’’ (Sfard et al. 1998, p. 43). Streefland (Sfard et al. 1998) asserted that mathematical discourse needs to allow for meta-cognitive shifts. More recently, Lee (2012a) noted the importance of using questions that encourage children to talk about their reasoning and their thinking approach. He termed this method thinking conversations and explained that the purpose is not to find out what children know or have learned or give them information, but rather ‘‘to elicit information on their thinking’’ (p. 7). It is a thinking conversations type of approach which was used in the data collection for this study. We attempted to use questions which require higher order thinking and are genuine questions which arise naturally in thinking conversations.

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Measurement Research A detailed search of the literature reveals that research projects have examined children’s knowledge of measurement attributes of length, area and volume (Barrett et al. 2011; Clarke et al. 2003; McDonough and Sullivan 2011; Outhred and Mitchelmore 2000). However, little research appears to have addressed early concepts of mass measurement. Denmark et al. (1976) reported a teaching experiment with seven Year 1 students. The purpose of the study was to develop students’ concept of equality as a relationship and to study the effects of instruction on addition, extended notation, and algorithms. Denmark used specially adapted school balance scales as a teaching tool. While the study was seminal work experimenting with concepts of early equivalence using balance scales, it took for granted the effective use of the tool with young children. Other research on mass measurement has shown that children in the early years of school have informal knowledge of mass (MacDonald 2010), perhaps developed during outdoor play activities prior to formal schooling (Lee 2012b) or from handling or weighing things at home (Spinillo and Batista 2009). Young children can compare masses by holding one object in each hand (Cheeseman et al. 2011) and can order objects according to their masses (Brainerd 1974). Children also have some informal understanding of pan balance scales (Cheeseman et al. 2014) and such informal knowledge can be formalised in the early years of school. For example, in a teaching experiment where 119 children in five early years classrooms were given challenging experiences in an investigations-based environment over a 1 week period, the majority of the 6–8-year-olds moved from using non-standard units to using standard units and instruments for measuring mass (Cheeseman et al. 2014). The children also showed diversity and complexity in their thinking during the classroom experiences where they explored and experimented, shared thoughts and theories, and many challenged each other in their learning (McDonough et al. 2012, 2013). In addition, classroom observations by Cheeseman and McDonough (2016) found that it is possible to stimulate curiosity about mathematics by: • • •

offering children challenging tasks and interesting mathematical tools; listening carefully to be aware of the learning potential children bring to the task; and encouraging curiosity by showing a real interest in children’s investigations.

In the research reported in this paper we were conscious of stimulating the curiosity of the participants using each of these techniques.

Method Twelve children aged 5–7 years old in their first and second years of formal schooling at a suburban government school in Victoria, Australia, undertook the exploration in three small group settings. Ethics consent was approved by the

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universities of the authors. Consent was then obtained from the children, the parents or caregivers, the teacher, and the school administration. The data were gathered on one day towards the end of the school year in a quiet music room where the equipment could be set up and the children’s explorations were uninterrupted. The suspended balances were a simple construction home-made with doweling, bulldog clips, string, and plastic pans (Fig. 1). There were enough balances for each child to have one. A range of equipment was available with which the children could experiment—small soft toys, commercial cubes (centicubes and unifix), counters, and sets of weights of several different kinds (metal, plastic cuboids, and plastic disks). We explained to the children that we were ‘‘very interested in how children your age think about things’’. The equipment was available on a large table and the children were invited to explore. Data were gathered for this study by observing children as they explored materials. The role of the adults was: to observe, to listen, and to interact appropriately; to probe children’s thinking; and to seek explanations of their actions. The researchers had informal conversations with the children while they investigated the equipment. These thinking conversations were characterised by features similar to those listed by Lee (2012a): children were respected as experts in their thinking and learning, adults questioned and listened, and the questions were intended to gain information about children’s thinking and to understand children’s thinking processes. We were trying to get children’s perspectives and their reasoning (Dockett and Perry 2007). Questioning and listening were important features of this research. The questions posed were ones that required higher order thinking—skills involving analysis, evaluation and synthesis and the creation of new knowledge (Mason 2010). Questions were often open-ended such as ‘‘How do you know …?’’, ‘‘What are you noticing?’’, ‘‘Why do you think that happened?’’ The answers framed in response required careful listening (Neumann 2014). Davies and Walker (2007) described three different types of listening: evaluative listening where the teacher was listening for a particular mathematical explanation, interpretive listening where the questions posed were information-seeking and often elicited some sort of demonstration or explanation, and hermeneutic listening which Fig. 1 The equipment with which to experiment

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was negotiated and participatory. This third type of listening describes the teacher as a participant in the exploration of a piece of mathematics where understandings emerged that were unlikely to have arisen had the learners not been given the opportunity to interact. They were characterised as complex and dynamic interactions where learning was seen as a social process and the adult’s role was one of participating, interpreting, transforming and interrogating. It is such hermeneutic listening that was seen as most important for this study. In the present study records of events were captured by two video cameras on tripods focused on pairs of chairs on opposite sides of the table. The purpose of the video was to document the actions and words of the children. All of the recordings were digital, enabling computer handling of all of the data. The video was examined for analysis and transcription. Video data were considered by Pera¨kyla¨ (2005) as a source of talk and text for analysis. Like all data sources, video records have weaknesses and strengths. Their weakness is that they ‘‘systematically miss the experience of the participants being recorded’’ (Hall 2000, p. 658). The video record offers a perspective that no participant could have had. The strength of video is that the records are plastic in the way that experience is not; they can be slowed down and watched repeatedly. They can contain things that no single participant could see and hear and they provide a record that allows account to be taken of actions. Cobb (1995) described analysis of data that ‘‘involved a continual movement between particular episodes and potentially general conjectures’’ (p. 35). This description resonates with the process used in this study. The iterative process used here involved many viewings of the data, alternating between a ‘‘close up’’ focus and a ‘‘standing back’’ to gain some perspective on the data, and raising and testing conjectures about the findings. Lampert (1990) characterised the ‘‘zig-zag’’ path of mathematical activity from conjectures to proofs by ‘‘revising conclusions and revising assumptions in the process of coming to know’’ (p. 30). At the end of the observation time the researchers discussed the children’s explorations and any events which were noteworthy at a round-table review. This enabled us to share our observations and our thoughts while they were still fresh in our minds. The video was later analysed in order to report the actions and thinking of individual children.

Findings and Discussion Findings presented here centre on five case studies where each has been chosen to illustrate the ways in which the actions of an individual child raised broad issues for discussion. Some reference to explorations by others of the 12 participants is included where deeper insights for discussion are facilitated. Creative Mathematics and Figuring Out the Tool: Thomas (5 Years) Thomas was observed using elements of imaginative play involving the scales. He said, ‘‘I’m making this a hot air balloon ride’’. At one stage he filled a pan with

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unifix and referred to the cubes as ‘‘the people’’. Perhaps the strings and plastic tubs looked to him like the ropes and baskets of hot air balloons. A researcher asked him about his play: Researcher: So what have you found Thomas? Thomas: This one would go higher and this one would go lower. Researcher: So what makes it go lower? Thomas: Because it has got more. From this reply we see that, while the experimentation was imaginative, it was seen by Thomas for the mathematics it was. Van Oers (2014) described productive mathematising as ‘‘dialogic, inquisitive, productive thinking’’ (p. 112). Typically mathematising is driven by personal engagement with a personal query that requires creativity and endurance. Van Oers linked mathematising to play where mathematics emerges from outside as an attribution of mathematical meaning to children’s actions or utterances by more knowledgeable others. As can be seen in the example of the hot air balloon it was the adult who explicitly stimulated mathematising by the child. The tools made Thomas inquisitive and engaged; he became creative about what he could do and what he could imagine. In all, he played with the equipment for more than 20 min changing his successive challenges as he solved each problem he posed for himself. Thomas was very interested in the suspended balance but he found it awkward to hold up the scales. He asked a researcher to hold the scales and later invented a way to suspend the balance from a video tripod (Fig. 2). He then spent some time getting

Fig. 2 Thomas using a tripod to hang his balance

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the pans in equilibrium. He became fascinated with adding more balances to his display until it began to look like an elaborate mobile. Thomas showed a range of thinking when using the suspension scales. He did what we have seen many children of 5–6 years old do; he alternately placed objects in each pan and watched the resulting swing of the pans as they see-sawed backwards and forwards with the weight. While we do not completely understand this behaviour, we hypothesise that the child is watching and observing the action and reaction as the pans are utilised. Often the child says ‘‘that’s heavy’’ as the pan goes down and it seems to reinforce the concept that the mass takes the pan lower. Thomas also tried all sorts of combinations of toys in multiples on either side of the balance scales and reported on his findings in terms of comparative weights. Having played around with the equipment he changed his purpose to finding two sides the same. We conjecture that this process is a pattern of exploration which is common to many children who have the opportunity to play with balance scales. We have observed it on many occasions but it is not found in the research literature as far as we can discern. Experience and Communication: Holly (5 Years) Holly was interested to experiment and as she held the balance in the air Holly remarked: Holly: It is like a scale but it’s not. Researcher: What is the same and what is different? Holly: It is like a scale ‘cos it measures weight. Researcher: I see. This observation shows that Holly was able to identify the common characteristics of a tool that was familiar to her. She classified the new home-made instrument as a weighing device. In addition she was able to transfer her knowledge of balances to this new context. Holly was putting objects in either side of her balance alternately and watching the pans swing and see-saw, the following exchange occurred: Researcher: What do you notice? Holly: This side is heavy (incorrectly pointing to the pan that was high in the air). Researcher: What do you think would happen if you added more to this (pointing to the higher) pan? Holly: It would go down. From this prediction we see that Holly has a concept of the consequences of her action. She was asked to try it to see and of course the pan did go down but not as far as the other side of the scale. Researcher: What happened? Holly: It is heavier. Researcher: Heavier because you put more in it?

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Holly: It is heavier. This exchange together with a later conversation with Adrian made us aware of the ways we use the English word heavier. It occurs to us that it may be useful to consider the different meanings the word holds: 1.

2.

Heavier can be a comparative term describing the change of mass of an object. I can, for example, be heavier than I was last week as shown on the bathroom scales. Heavier can also be a comparative term between two different objects. For example, Andrea’s school bag is heavier than mine.

Listening carefully to children trying to express their mathematical thinking about the comparison and measurement of mass, made us conscious of situations where it is not clear to children (and to us) which meaning of heavier is being used. Of course it is not the only mathematical comparative term to be troubling in this way. Long ago Herbert Ginsburg (Skemp 1986) was filmed interviewing a young child about who had ‘‘more’’ counters. The lower parts of two glasses were masked from direct view behind a stack of books. Simultaneously a red and a black counter were dropped into the glasses seven times. The child is asked ‘‘Which has more or are they the same?’’ The child said, ‘‘They are the same.’’ One counter was added to the black and two further red counters were dropped into the glasses out of direct sight. The question was then asked of the child ‘‘Which has more?’’ The child responded, ‘‘The red one has more.’’ Ginsburg then said, ‘‘But I did put something in the black one didn’t I? Doesn’t it have more?’’ (Alluding to the fact that some counters had been added to his collection). The child responded ‘‘Yes, but the red one has more.’’ This response indicated that the child could think simultaneously about more as an increase and more as a comparison. The mass measurement equivalent statement in the case of what Holly was doing might be something like, ‘‘I have made one side heavier but it is still not as heavy as the other pan.’’ The responses of the children observed in this study raise the need to be more careful or specific in the way we use comparative terms. We recommend that we use the phrase heavier than in teachers’ work with children. Our teacher–researcher, Dianne, reflected on the experience independently and wrote: I’m fascinated by the unique language that we use for mass measurement. I feel more strongly now that children need many opportunities to explore with the materials, engage in meaningful conversations with their peers and teachers about the meaning of the words – weigh, heavy, light, weighing, and weight (wait). I have not considered the specific vocabulary necessary to develop these measurement skills before. This reflection highlights the fundamental importance of language in communicating mathematical thinking and that children need to acquire particular vocabulary to communicate. As Sfard (2001) noted, communication is thinking.

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Mathematical Reasoning: Adrian (6 Years) Adrian could heft two toys to judge which felt heavier on his flattened hands (Fig. 3). The toys had a difference in mass of 11 g. He said maybe the bear was heavier, then when he was encouraged to put the same toys in the balance scales he smiled and giggled to see the result—as if the scales confirmed the matter for him. This exchange followed: Researcher: What do you think? Adrian: The bear is heavy and the fish is light. Researcher: How do you know? Adrian: Bears sometimes are heavy. It is quite startling to observe children who seem to be following a logical line of deductive reasoning to interpret the results of a comparison of weights, suddenly give an explanation which is a non sequitur—an argument in which its conclusion does not follow from its premises. We have met responses like these before and have come to think it is more to do with children’s emerging powers of mathematical argumentation rather than their innate reasoning. Adrian estimated by hand and had a slightly uncertain tone when he said the teddy was heavier. However, as his smile and chuckle revealed, he was happy to see the scale tip down with the teddy. In his mind his prediction had proved correct. It was the explanation and his justification that let him down in this case we think. We expect a mathematical justification whereas children can consider a ‘‘logic of what I know’’ to be perfectly adequate. Salgad and Stacey (2014) called this type of justification ‘‘drawing on personal information’’ where students ‘‘think and argue in personal terms rather than mathematically’’ (p. 61). Yackel and Cobb (1996) also found that when asked to explain their thinking, students initially gave explanations with social Fig. 3 Adrian discriminating between two toys with a difference in mass of 11 g

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rather than mathematical bases. However, they noted that participation in inquiry mathematics facilitated progression in students’ explanations. Providing experience in mathematical investigations with the expectation that young children can reason and justify their reasoning is a sensible course of action for teachers. Cheeseman (2008) found that ‘‘interactions that challenge children to think about their mathematical understandings are a critical factor in their learning … many young children spontaneously remember these conversations and can reconstruct their thinking’’ (p. 296) and a large proportion of 5–7-year-old children (42%) could reconstruct and justify their thinking. Dialogue with young learners is important also to avoid adults making assumptions about children’s interpretations or meanings (McDonough 2002). The case of Adrian reminds us of the importance of eliciting children’s mathematical reasoning and of not assuming what children might be thinking. Noticing: Miles (Aged 5) Miles made an initial comment which indicated that he was very aware of the purpose of the tool, ‘‘I know you put stuff in it and you see if it’s heavier or not’’. He gave a description of how scales work saying, ‘‘When you put more in, one gets lower and one gets higher. This is getting lower and lower because it’s got so much in it.’’ This idea of ‘‘much’’ is one which made us stop to think. In this example we agreed that Miles meant the stuff, quantity of matter, bulk, or material. On the other hand Adrian used the word in a different way. Adrian had his balance scales at a point of equilibrium when he was investigating plastic weights. The set of materials is built as proportional masses as shown in Fig. 4. Adrian had yellow masses in one pan and red masses in the other. Researcher: What have you found Adrian? Adrian: They are the same but they’re not the same much (sic). The conversation continued as we tried to interpret ‘‘not the same much’’. In hindsight it is clear to us that Adrian was noticing that the total masses were equivalent yet the number of pieces needed to make each total mass was different. He was attending to weight and number simultaneously. This idea of equal masses being composed of different numbers of objects is of course the central idea of mathematical equivalence. This idea was evidenced by Ben. ‘‘Big’’ Mathematical Ideas: Ben (6 Years) It was clear that Ben came to the session with a working knowledge of balances. He said, ‘‘You have to make them equal’’ when further pressed he said, ‘‘They are for Fig. 4 Commercial plastic masses made as proportional models

red (5g)

blue (10g)

yellow (20g)

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weighing things.’’ He knew how to work the scales and the purpose for which you might use them. Ben was in his first year of formal schooling and when we later spoke to his teacher, she said that they had done a ‘‘little bit’’ of weighing with commercial pan balances earlier in the year. Of course Ben may also have had experiences with weighing at home and outside school. Ben was interested in experimenting and paid attention to how we had made the sets of suspension balances. He immediately saw the similarity between the pan balances on stands which had been used in his classroom and the balances we had made. He generalised the suspension balance as a type of balance which he knew how to use. He set about investigating how accurate they were. In comparing two soft toys it became apparent that to judge which was heavier Ben was looking at the top of the plastic pans and comparing them horizontally. We have noted in the past that children are sometimes distracted by objects sticking up out of the top of pans but this was not the case with Ben. Other children have been observed attending to the beam between the pans, the distances between the bottom of the pans and the table-top, and whether the pointer/indicator on a commercial balance scale is vertical. We think it is important to know exactly what each child is noticing as they interpret balances as some children are distracted by irrelevant things. Later in his exploration Ben told us that when two cubes are put in each pan the scales show they are ‘‘the same’’. In an effort to have him expand on the idea he was asked, ‘‘But the cubes are not the same colour’’. Ben looked completely confused as if thinking ‘‘what has colour got to do with it?’’ He was clearly focused on the numerical equivalence of the cubes in the pans of the scale. As if to convince us he demonstrated adding one to each pan saying, ‘‘Three here is the same as three on this side, see?’’ He was attending only to the number of cubes and, for Ben, the numerosity of the mass of cubes was a convincing argument. Other children also demonstrated ideas of equivalence in their explorations: Maddie said, ‘‘We need this one to be going down, so that it’s equal.’’ Sam and Adrian found that ‘‘the small metal weight equalled twenty grams.’’ Ruby and Lily said that they found ‘‘none of them equal’’ as they used the small metal weights. Marisa and Zander were jubilant, ‘‘Thirty-seven unifix and thirty-seven unifix are they the same? Yes they are! Another exact! Exact three times.’’ As a result of these and other observations we hypothesise that balance scales may be a useful tool to help children to build concepts of numerical equivalence. Ben experimented further with equivalent relationships when he had equal numbers of unifix cubes on the scales and he simultaneously added blue plastic masses to the pans. Ben described what he was doing, ‘‘I’ve got it even, so when I have got the same amount in each hand, I add the same to each side and it stays even’’. He had mixed the media and was considering adding equivalent amounts to keep the scales in equilibrium. This mathematical generalisation—when you add equally to equivalent quantities, they remain equivalent—shows quite a sophisticated understanding for a child of 6 years old. Ben brought his knowledge of balances to his experimentation as he used the new tool to investigate numerical equivalence and was able to generalise his mathematical thinking. As we were packing away Ben asked whether he could

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take the balances back to his classroom to show the children what he had been doing as he was keen to share his thinking. The story of Ben reminds us of the possibility of ‘‘little’’ children dealing with ‘‘big’’ mathematical ideas: transfer of knowledge, equivalence, generalisation, and the importance of the emotional or affective engagement in mathematical investigation.

In Conclusion This study emphasises the importance of the quality of the thinking conversations we conduct with young children to elicit thinking and to stimulate mathematising. Taking a social constructivist perspective allowed us to listen and to come to know students’ current and developing understandings about the measurement of mass. Our insights went beyond mathematical content to student actions, sometimes described as proficiencies (e.g., ACARA 2014). Young children were seen to be able to describe their experiences, to reason, and to justify their thinking. The study demonstrated how important mathematical concepts such as equivalence can be experienced and understood by young children. In addition mathematical generalisations can be made by children when they are encouraged to state what is happening in general terms. The role of activities that stimulate curiosity and experimentation was found to be important in creating inquisitive and engaged children who persist with challenges they pose for themselves. The research has several implications for classroom teachers who may consider: •

•

• •

•

•

Creating the opportunity for young children to play with mathematical tools to investigate how they work and to pose challenges for themselves with the equipment; Encouraging children to reflect on their actions with materials so as to actively construct knowledge and build mathematical meaning (Clements 1997; Sarama and Clements 2009); Expecting that young children can reason about mathematical investigations and justify their reasoning; Watching children’s behaviour as they experiment as this can help to reveal their mathematical thinking. Teachers can then hypothesise about what the child is learning and how the child is testing conjectures; Listening carefully to children expressing their ideas as this made us conscious of the fundamental importance of facility with language and reinforced to us that communication is thinking; and Eliciting children’s mathematical reasoning and not assuming children’s thinking.

This small study led us to consider deeply ways to have thinking conversations with young children and to learn from listening and observing them. We were delighted to find that simple home-made tools made children inquisitive and creative problem-posers. We were reminded of the power of investigative play to

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provide opportunities for children to mathematise the world. We hope that in reporting our observations of young children we have emphasised the importance of attending to what children are noticing and of eliciting children’s mathematical reasoning. As can be seen from the research presented here young children can indeed think about ‘‘big’’ ideas of mathematics in a natural and meaningful way.

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