# Thirty-one is a lot! Assessing four-year-old children's number knowledge during an open-ended activity.

IntroductionChildren are born with innate mathematical abilities and have the capacity to engage in 'significant mathematical thinking and learning' from a very young age (Clements & Sarama, 2009, p. 3). Early mathematical thinking may begin with comparisons of set sizes; these comparisons may be 'the foundation for basic mathematical ideas that are essential to an understanding of number' (Saracho & Spodek, 2008, pp. 21-2). This exploration of quantity occurs without necessarily counting or knowing the total sum of the amounts. Certainly, children demonstrate a wide range of mathematical thinking and have their own understandings of 'preschool arithmetic' (Clements & Sarama, 2009; Doig, 2005; Vygotsky, 1978).

Assessment is 'the process of observing children in everyday experiences, analysing those observations and recording the information' (ACECQA, 2011, p. 1). Numerals and quantity are one of the earliest mathematical concepts children learn (Clements & Sarama, 2009). Consequently, assessing children's knowledge of number symbols and their relationships with quantity is a critical first step if early childhood educators are to address the gap between what we know about young children's mathematical understandings and how we enact practice in the early childhood setting (Ginsburg, Jacobs & Lopez, 1998).

Early childhood education in Australia has undergone a significant change and there is now an unequivocal emphasis on the quality of educational programs provided in early childhood settings across the country (ACECQA, 2011, DEEWR, 2009). The Early Years Learning Framework for Australia (EYLF, DEEWR, 2009) sets out a range of learning outcomes, including numeracy-related concepts and practices. The EYLF also states that observations and assessment of children's learning and development should directly inform the teaching and learning cycle. Yet research suggests that there is a gap between how we teach children mathematical concepts and how we assess children's mathematical understanding (Baroody, 1989; Fleer, 2008; MacDonald, 2009; Tishman & Palmer, 2005). Authentic assessments of children's mathematical understanding in the early years are the starting point for understanding what to teach and how to teach it (Clements & Sarama, 2009; Klibanoff, Levine, Huttenlocher, Vasilyeva & Hedges, 2006; Stake, 2010; Sun Lee & Ginsburg, 2009). In other words, if educators are unsure how to assess children's understanding and skills, then effective teaching of mathematical concepts is unlikely to be responsive to children's academic needs.

This paper is concerned with how early childhood practitioners can authentically assess children's understanding and knowledge of number in a play-based program. To this end, we begin by providing a brief review of key issues in assessment in early childhood mathematics, and to consider how varying approaches to assessment can capture children's capacity to make associations between number symbols, verbal numbers and the quantity assigned to numbers.

Assessment in early childhood mathematics

Children learn at different rates and in different ways. Assessment should accordingly be ongoing, take many aspects of learning into account, and be integrated in the learning program--to adapt planned learning experiences to better fit each child's interests and needs, and to evaluate the effectiveness of the program itself (Hughes, Gullo & Kindergarten Interest Forum, 2010). Assessment in early childhood needs to be personal, meaningful, real, relevant to children's everyday experiences, and informed by context (Carruthers & Worthington, 2006). Furthermore, mathematics assessment needs to be connected to the child's prior knowledge and grounded in social contexts such as play, to support the further learning of mathematical concepts and strategies (Carruthers & Worthington, 2006; Perry, Dockett, Harley & Hentschke, 2006; Perry & Dockett, 2002; Sun Lee & Ginsburg, 2009). Authentic assessment builds on what children already know (Fleer, 2008), and provides ways for children to express their understandings in ways that are meaningful to them (Clements & Sarama, 2009).

In recent years, there has been an increasing interest in developing ways to engage young children in developmentally appropriate mathematics activities and assessment that are embedded in a play-based curriculum (Carruthers and Worthington, 2006; MacDonald, 2012). This interest has emerged in response to the field's growing recognition of the importance of assessment and its impact on cycles of planning, informing curriculum and guiding teaching practice (DEEWR, 2009). Unlike summative assessment, formative assessment is ongoing and includes cycles of teaching, learning and planning:

Each child's learning and development is assessed as part of an ongoing cycle of planning, documenting and evaluation. It is an interactive process that drives development of the program (ACECQA, 2011, p. 32).

Formative assessment helps the educator to understand children's learning processes and to identify their strengths, skills and understandings (DEEWR, 2009), as children may demonstrate capabilities that exceed or do not yet meet their assumed level of understanding (Walsh, 2013).

Despite the importance of assessing children's understanding and skills, there has been little discussion to date about how to enact accurate, formative assessment within a play-based curriculum for preschool children. The diversity of children's mathematical skills needs formative assessment strategies that are open-ended, because children express meaning using a variety of creative, inventive and sophisticated processes (Edwards, Gandini & Forman, 2011; Gardener, 2011; Wright, 2011). Traditional approaches to assessment that include closed questions and interview format strategies are less productive for preschool children and may not provide sufficient scope for children to communicate non-verbally or to express connections to existing knowledge in ways that are meaningful, based on the child's own experiences, existing knowledge, contexts, cultures and ways of being in the world (Wright, 2012; Carruthers & Worthington, 2006).

Standardised assessments often test one area of mathematical understanding in isolation and may not provide for the interrelated and creative ways in which children understand and represent numerals and quantity (Bobis, 2008; Smith & MacDonald, 2009). Modes of expression typically employed by young children include aural and musical, kinaesthetic, visual and spatial, all of which contribute to communication (Wright, 2012). While drawing, children make sounds, talk, gesture, move, incorporate objects in their work, use their bodies to act out an aspect of their work and make references to others' work. Children draw on these processes to create meaning from 'lots of different stuff' as it is a way to communicate and make meaning before learning to write (Kress, 1997, p. 7).

There is a renewed interest in children's mark-making as a platform for assessment in general (Carruthers & Worthington, 2006; MacDonald, 2012; Wright, 2012), and in particular as a way to assess children's mathematical understanding (Carruthers & Worthington, 2006; MacDonald, 2012). Creating opportunities for children to express their mathematical understanding enables educators to understand children's reasoning and to facilitate children's application of thinking strategies to new mathematical concepts, creating a strong foundation of mathematical knowledge on which to build (Kilpatrick, Swafford & Findell, 2001; Perry & Dockett, 2002; Thomson, Rowe, Underwood & Peck, 2005).

However, evidence suggests that children are typically not provided with sufficient opportunities to build on the wide range of existing mathematical abilities and knowledge they bring to the early childhood setting (Carruthers & Worthington, 2006; Ginsburg & Ertle, 2008) and instructional support enacted in Australian early childhood settings has been found to be quite low (Tayler, Ishimine, Cleveland, Cloney & Thorpe, 2013). Higher quality instructional support assists children to integrate new mathematical knowledge with existing knowledge, ensuring that mathematical concepts continue to hold meaning and relevance for each child (NAEYC & NCTM, 2002). A critical first step to achieving high-quality teaching about numbers and quantity is to assess what children already know.

Study design

This study is interested in both the product and the process of children's work. The narrative inquiry approach (Chase, 2005; Clandinin, 2007) taken is aligned with the study's focus on the narration of the experience as it unfolds and the final product--the child's drawing in this case (Kramp, 2009). The children's descriptions of what they are doing and why they are going about it in a particular way is of primary interest as it is at this point that the teacher observes and analyses the mathematical reasoning reflected in the children's discourse and work as their thinking evolves, is constructed and is represented (Wright, 2012, MacDonald, 2013). Consequently, analysis must also be flexible and endeavour to include the scope of children's responses to the assessment activity, including both the processes and outcomes of mark-making and sustained conversation (Carruthers & Worthington, 2006; MacDonald, 2013; MacDonald & Lowrie, 2011). By analysing the product and the narrative together, each artefact provides evidence of 'what children know and understand, based on what they make, write, draw, say and do' (VEYLDF 2009, p. 13).

This approach provides opportunities for verbal and nonverbal modes of communication that are integral to the meaning-making processes (Carruthers & Worthington, 2006; Kramp, 2009; MacDonald, 2013; Siraj-Blatchford, 2009; Wright, 2011), and allows for the experiences of an individual to emerge (Creswell, 2008). The open-ended, conversational nature of the activity in the familiar environment of the early childhood setting provides time for children to see, discover and share understandings that might otherwise be missed or not communicated (Cohen, Manion & Morrison, 2010). Drawing captures 'the process of constructing a mathematical concept or relationship' (MacDonald & Lowrie, 2011, p. 40) and enables children to record and share their thoughts. The combination of children's narratives and representations of mathematical ideas enables the observer to capture the processes involved as children construct ideas about mathematics as they unfold (MacDonald, 2013). Essentially, children's mark-making and accompanying narratives provide an insight into their mathematical reasoning, facilitating authentic assessment.

Procedure

The study was conducted at three early childhood centres in metropolitan Melbourne over a three-week period. Each centre offered a play-based curriculum and reported being influenced by the Reggio Emilia approach (Edwards et al., 2011). The philosophical approaches of the centres supported the study method, as the children were familiar with working alongside educators and discussing their ideas. Forty-seven children, age ranging from 4.1 to 5.7 years, participated in the study across the three centres. Centre directors, teachers and parents all provided informed consent to the study, and children provided (ongoing) assent by agreeing to participate after the researcher had explained what they were being invited to do.

A selection of ten wooden numerals (0 to 9) and a range of drawing materials were set up as an open-ended, small group activity (see Figure 1). The space was organised so that small groups of children could draw and discuss their work together. Children were invited to select a numeral to trace around. It quickly became apparent that some children wished to draw two- or three-digit numbers and so a second set of ten wooden numerals (0 to 9) was added to the first set.

The first author asked each child to draw 'how many' the numeral(s) they had selected represented to them and engaged in conversations with the children about their work, taking handwritten notes of remarks made by the children as well as general observations about how the representation of numerals and quantity took place. The conversations between peers were documented when they demonstrated evidence of children's numerical skills and understandings.

The following prompts were used to sustain conversations with children about their representations of 'how many' the numerals meant to them:

* Is there anything you would like to show/tell me about the number you have chosen?

* Would you like to draw me a picture of how many the number (5) is?

* How do we know it is (5)?

* Would you like to show/tell me about the number (5)?

* Would you like to tell me about your drawing?

* I wonder ... what do you think would happen if ...

Asking open-ended questions was a purposeful strategy as it took the focus away from getting it 'right' and tried to evoke more detailed and in-depth responses. They enabled the researcher to follow the children's lead and 'go with the flow of children's ideas', supporting increased engagement with the activity, providing opportunities for children to talk about their work (Pianta, La Paro & Hamre, 2008), facilitating sustained shared thinking, an interaction during which two or more individuals collaborate intellectually to solve a problem or clarify a concept (Siraj-Blatchford, 2005, p. 1). Children's responses to the questions enabled the observer to assess the accuracy of the connections between numerals and the associated representations of quantity. Encouraging the children to talk about their work while they were drawing supported higher order thinking and extended responses to questions, providing greater insights into children's thinking processes (Barden, 1995; Walsh, 2013) and enabling the observer to challenge children's thinking.

Children took 7-12 minutes to complete the activity once. On completion, a photograph was taken of each child's work for inclusion in the data set. Later, the handwritten observations were paired with corresponding photographs then analysed for evidence of accuracy of representation of the selected numeral, numerical understanding using the developmental progression points of The Learning Trajectories Approach (Clements & Sarama, 2009), and mastery of the counting principles (Gelman & Gallistel, 1978). Children's spontaneous use of mathematical symbols, references to their own ages or the ages of family members or friends, their use of gesture or movement to demonstrate 'how many' were also noted. By analysing the product and the narrative together, each artefact provides evidence of 'what children know and understand, based on what they make, write, draw, say and do' (VEYLDF, 2009, p. 13).

Findings

The wooden numerals provided a focus for mathematical discussions and were an effective way to engage children's interest in, and focus on numbers. The ways in which children were observed to use the wooden numerals varied (see Figure 2) and frequently, more than one strategy was employed.

Each child engaged with the activity when invited and conversed with the researcher during the activity. The researcher (first author) collecting data is an experienced early childhood educator, but did not have extended contact with the children. Despite not having an existing relationship, all the children participating in the study were willing to join the small group activity and freely offered their commentary as work progressed (see examples in extracts below). Thirty-eight children discussed their numeral(s) with one or more peers, 28 children interacted with one or more peers while counting and 28 interacted with one or more peers while representing the quantity of the selected numeral. The activity, designed to facilitate assessment, was highly effective in creating opportunities for 'maths talk' (Klibanoff et al., 2006; Cohrssen, Church, Ishimine & Tayler, 2013) and peer scaffolding.

When asked to draw 'how many' the numeral represented, children provided highly individualised responses that included for example counting on fingers, using blocks as counters, referring to their own age and the age of others, comparing and drawing objects and symbols, referring to street numbers, and demonstrating number beats with music and gesture. At times, 'how many' was expressed as length, height, distance, time, size, capacity and speed demonstrating the children's dynamic and interrelated understanding of mathematical concepts and highlighting both the range of ideas children bring to mathematical thinking, and children's creative capacity for expressing their mathematical reasoning.

Data provided by the children, in the form of drawings and accompanying narrative, reflected multiple strategies for representing numbers and demonstrated links to children's lived experience. Four key and frequently occurring characteristics of children's work are illustrated and discussed below.

Spontaneous physical movement, using gesture and concrete objects

Forty-two children used gesture and movement in conjunction with their drawings to represent quantity. Thirty-two used gesture and movement to explain height or distance, as well as the concept of zero. Highlighting the importance of assessment taking place in a familiar environment where children can move freely and select resources to support their own learning, Charlie (4.6 years) demonstrates, in the example that follows, his understanding of '31' by sourcing Lego blocks to move and group.

Charlie looked through the wooden numerals and stated that he was 'going to do it in big order and then smaller order'. He placed the numerals from 10 to 1 on the table, verbally counting backwards without errors. He looked at the wooden 1 and compared it with the 7 before suggesting that we play a game in which the numbers are mixed up and we have to put two together, 'like 44'. Charlie made 11 and then 55. He continued to explore the numerals and then carefully traced the numerals to make 31 (see Figure 3).

Researcher: How do you know this is 31?

Charlie: Because the 3 is 1st and the 1 is 2nd.

Researcher: Could you draw a picture for me to show how many 31 is?

Charlie: No--it is just as it is. (He gives the number a big tick.)

Researcher: I wonder how many 31 is, could you think of a way to show me?

Charlie: It's a lot, I don't want to draw it, it will take a long, long, long, long time.

Researcher: Okay, sure. Well, maybe you could draw me how many just the 3 is, do you think that would take less time?

Charlie: Sure. (He draws 3 more ticks, counting as he draws 1, 2, 3.) This is 3, not including this tick.

Researcher: So if this is 3, how many more groups of 3 would I need to make 30?

Charlie: Lots more.

Researcher: I wonder how many?

Charlie: I will show you. (Charlie stands up, looks around, goes and collects some Lego blocks. He starts counting out 31 blocks, grouping them in sets of 3. He runs to fetch more blocks as he needs them.) See!

Researcher: You were right. Thirty-one is a lot.

Charlie: Let's play another game.

Researcher: Okay, sure. But first can you tell me how many groups of 3 make up 31? I am wondering how many it is, it looks like a lot

Charlie: Okay, hang on, let's work it out ... (Charlie counts out 10 sets of three by moving first each set to one side, and then moves the one left over). Okay, there is 10 and this one here.

Researcher: Wow, that is a lot. How many is that altogether, I wonder?

Charlie: Um. Okay, hang on ... (Charlie audibly counts each block in each set up to 30. He comes to the single block and says) And one more makes 31, all of these are 31.

During this activity Charlie stood up to work, walked and ran back and forth between the Lego container and the table, carried and sorted the blocks for counting, drew ticks to represent 3, tapped out the ticks when counting, moved each set of blocks when counting out the 10 sets and 1 more for 31, and individually tapped each block when counting 31 in total. Later the same day, Charlie drew 31 circles in a notebook to show 'how many altogether (see Figure 4), demonstrating the extent to which this activity promoted ongoing reflection. He wrote 31, reversing the numeral 3, and then encircled the 31, stating that this is 'the wrong way' to write it.

The data demonstrates that Charlie has mastered the counting principles. He used counting all and counting on strategies, and accurately grouped and described the Lego blocks as 10 sets of three with one left over. Charlie independently corrected the direction of the 3 in his work (lower right hand corner of Figure 3) and by ticking his work, demonstrated awareness of a symbol typically used in formal school-based education for 'getting it right'.

It is interesting to note that nine of the 47 children used symbols such as +, - and = to explain their counting strategies. Of these nine children, seven used mathematical symbols to add sets together and calculate the sum. Twenty-five children referred to 'more than', 'less than', and 'counting one more', suggesting that they were familiar with mathematical concepts and language and ready to explore the symbols for addition and subtraction (Clements & Sarama, 2009).

Pictographic representations and verbal references to real life

The data revealed that children's understandings of numerals and quantity related to their personal, social and cultural contexts and were grounded in everyday understandings. Thirty-five children referred to their own lived experiences while counting, 27 did so while representing the quantity of the wooden numeral, and 42 children referred to their own lives when discussing the numeral with the researcher or with peers. Examples of these references to their lives included the street numbers of their homes, birthdays, and references to personal possessions or experiences. For example, when asked 'how many' the numeral five represented, 15 children responded 'I am five', as though they embodied the quantity. A further nine children referred to the age of a sibling. Three children allocated ages to the figures in their drawings. In this way, numbers were a way to count and compare 'bigness' and 'how many' thus served as a measure of age. Street numbers were described as showing 'how many your house is', and 'how many your house is now that you have moved' and consequently counting the difference between the house numbers tells us 'how many houses you have moved away'.

Alice (4.9 years) drew a row of flowers (see Figure 5) to represent the 'total amount' but then explained that it can also represent 'volume', from smallest or softest to largest or loudest as it reminded Alice of the symbol for volume on her computer.

Tom incorporated a combination of numbers, people, Star Wars[TM] figures, a car stuck in traffic, a rocket launch, letters and a self-portrait to represent his understandings of numerals and quantity (Figure 6). Throughout his narration of his work, Tom used mathematical language to describe the elements of his work using comparative terms such as 'larger than', 'shorter', 'faster', 'heaviest', 'the oldest' and finally, 'running late'.

As he was working, the tail of a creature he drew reminded Tom of an 8, prompting him to write 8 beside it. He tapped eight figures in his work and counted to eight using one-to-one correspondence.

Tom practised writing the numeral 5 three times before shouting out, 'I can do it!' When asked 'how many' the numeral 5 represented, Tom replied that he is 'that many' and drew a self-portrait. As the conversation continued, Tom said that now he could 'make a 5', he wanted to make a card for his father's 50th birthday.

If Tom had created this work without the purposeful provocation of the wooden numerals and the perseverance on the part of the adult in encouraging both the child's sustained engagement with the activity and his narration of his meaning-making, it is likely that the product would have been the focus, rather than the process of producing the work. It is through Tom's detailed explanations that his recognition of the number symbols of 5, 8 and 50 could be identified.

Many children referred to zero, describing it as 'the crack between you and your body when you lie on the floor', 'nothing, but when you write it down, it is something', 'like an "O" as well', and 'even though you can see it (the numeral), it is still nothing'. One child provided a detailed explanation of zero, explaining that adding zeros to a number 'makes a number bigger, even when it is making it smaller because zero is nothing, so we are putting more nothing with the number'. Encouraging children to articulate their understanding of mathematical concepts provides the early childhood educator with an insight into the child's existing knowledge and points of reference.

Reversed numerals and writing right to left

Thirty-five children reversed one or more drawn, traced or written numerals and/or started work on the right-hand side of the page and proceeded towards the left. For the most part, children did not attend to the fact that their numerals were reversed. When another child talked about orientation, however, children changed the direction of numerals. Consider, for example, Alice and Lotte's work undertaken as they sat side by side and discussed their drawings (Figures 7 and 8):

Lotte: What is that? (Pointing to the 5 on Alice's drawing.)

Alice: That's a five.

Lotte: No, that's a two, see. (Lotte points to her representation of the numeral 2. Both look similar.)

Alice: (Picks up wooden 2 and compares it with the traced symbol in her drawing, then turns the two upside down.) Looks like five to me.

Lotte: (Places the wooden 5 and 2 side-by-side.) They are not the same.

Alice: No, only when you put it like this. (Turns the 2 upside down.)

After this discussion, Alice re-wrote the numerals above the traced (reversed) numerals (top right corner of Figure 7), having learnt about the spatial orientation of the numerals from her more knowledgeable peer. Six children were observed to correct reversed numerals after these conversations, demonstrating the importance of peer discussion and collaboration during mathematical tasks.

As reported by Johansson (2005), numeral reversal observed in this study did not impact upon children's understandings of quantity: even if a numeral was reversed, children named the numeral correctly and still represented quantity correctly. Bethany's (4.7 years) representation of the numeral 5 demonstrates this clearly. Bethany started her work by tracing the wooden numeral 5 in reverse. She then wrote five B's in an ordered row to show 'how many' 5 represents (see Figure 9).

Bethany narrated her work, saying: 'I will draw 5 Bs for my name (writes the third B and pauses). This looks like 8 ... (continues to write Bs) and now these are bees with wings on. Hang on ... (pauses then points with pencil] first is two "Bs" second is one "8" and next is two bees. (Counts and taps each item with her pencil) 1, 2, 3, 4, 5!' This monologue is rich with mathematical understanding. Bethany demonstrated that she understands the relationship between 5 and how many '5' represents even when the set contains different items--one being a number. The third letter B could also be a number 8, demonstrating recognition of the similarity between an 8 and a B. This example also demonstrates how children's thinking evolves as they draw and explore their understandings of these symbol systems.

Evidence suggests that if unaddressed, number reversal that persists into later primary years has an impact on arithmetic problem solving (Johansson, 2005). Bethany's work, however, demonstrated mastery of one-to-one correspondence, the ordinality and cardinality principles, and the abstraction rule. She also compared set sizes and added sets. The reversal of the 5 in her drawing did not appear to have any impact on her mathematical understanding but rather highlights the interrelated, complex, highly creative and inventive nature of children's mathematical thinking.

The complexity of children's understanding of number and representations of quantity

In the following conversation, Lily (5.3) and Sophia (4.9) are using numbers as units of age, they are counting on, measuring, comparing, estimating and ordering. Their discussion is an example of how children can explore all of these concepts, simultaneously. The observation takes place with Lily and Sophia sitting side by side. They are using the wooden numerals, discussing age and who is 'the biggest'.

Lily: (Tracing the wooden numeral 5.) Every time I have a birthday I am one more, now I am 5, but last time I was 4 ... next time I will be 6.

Sophie: (Tracing the wooden numeral 4.) That is a lot; that is big ...

Lily: My cousin is 9; she is this big. (Lily holds her hand to one side, indicating height.) My sister is 8 and she is this big (moves her hand slightly higher indicating that 8 is 'bigger' than 9). 8 is bigger than 9.

Lily counts one on and counts one back. She demonstrates logical and complex understandings of the relationships between age, number, quantity, size and measurement. However, while her sister may be bigger (or taller) than her cousin, eight is not more than nine, a teachable moment that the educator could use to clarify the value of each number, given the interactive design of this type of assessment.

Lily continues to share her understanding of 'how many' five is, by drawing five unicorns. During this process Lily provides the following narrative:

Lily: The big one is 25, the small one is one and a half, medium is teenager. (Lily then appears to note that two unicorns are the same size.) Two of them are teenager (Lily counts up to eleven tapping her pencil on the page for each count.) Two of these unicorns are eleven, that last one is two ... No, no not this one (pointing to a smaller unicorn on the page), that one is smaller, that one is two (points to the unicorn on the left).

Lily's narrative demonstrates 'evolving ideas' (Wright, 2012, p. 18) as she reflects on her understandings of age and size, number and size, comparison and ordering. We observe that Lily understands the cardinality rule, evidenced by her representations of five (drawing and counting five unicorns), and her counting and tapping to 11. Lily assigns ages to the unicorns based upon their size: smaller unicorns are younger. Her graphic representations of quantity thus enable an educator to assess her understandings of height and number sequence.

Through these various representations of number, children's meaning-making processes are revealed, which illuminate our understandings of the child's final artefact or product (MacDonald, 2013). Being actively engaged in children's processes of meaning-making as they emerged provided critical insights into their mathematical knowledge. It also highlights the interconnectedness and parallel acquisition of mathematical concepts.

Conclusions and implications for practice

This paper has detailed an early childhood mathematics assessment strategy which provides valuable insight into how children connect numerals and quantity, as well as providing additional insights into their mathematical thinking. The activity provided opportunities for children to demonstrate their knowledge of numerals and quantity in individual, verbal and non-verbal, multifaceted ways (Carruthers & Worthington, 2006). While this was a small study, it adds to a growing body of research into the effectiveness of assessing children's mathematical reasoning through the processes of mark-making and sustained conversation (Carruthers & Worthington, 2006; MacDonald, 2013; MacDonald & Lowrie, 2011).

Several conclusions may be drawn, each with clear implications for early childhood mathematics pedagogy:

* Children's mathematical reasoning is creative and inventive. Informal, formative assessment is critical if educators are to provide contingent learning experiences that both consolidate children's current understanding and keep pace with children's evolving understanding.

* Providing children with a range of resources with which to explore mathematical concepts supports their learning and encourages the transfer of knowledge from one context to another.

* Children's understandings of numerals and quantity are grounded in real, everyday experiences. Early childhood educators should encourage mathematical thinking and language in real, everyday experiences across all aspects of their programs. Formative assessment that takes place during the typical play-based room program provides a familiar setting for children to engage with materials used for assessment. The assessment strategy is developmentally appropriate and enables children to document and communicate their individual mathematical understandings.

* Children's understandings of number and quantity are interrelated, complex and unique. In order to assess children's understanding of mathematical concepts, educators need to ask open-ended questions and engage in sustained conversations that provide opportunities for children to take the lead. Interactions of this nature are best suited to small-group activities. By facilitating opportunities to assess children's knowledge, it becomes possible for educators to identify the mathematical concepts and strategies that children use for their mathematical reasoning. Collaborating with children during mathematical assessment helps educators to identify teachable moments as they emerge (Ginsburg & Ertle, 2008). This further informs intentional teaching strategies and supports ideas for planning effective learning experiences to meet the child's unique needs, while also building strong foundations for future mathematical learning.

* Small group activities provide opportunities for children to share their knowledge with their peers. Using this assessment strategy with small groups of children facilitated peer learning and the sharing of perspectives, providing insights into the ways in which children co-constructed mathematical learning. This assessment strategy promoted discussion about mathematical concepts and demonstrated that mathematics can be a social and engaging experience for children and educators, with positive assessment and learning outcomes.

* Children's representations of numerals alone do not always reflect their depth of mathematical knowledge. Evidence provided by children's narrative in conjunction with their drawings provided greater insights into their existing and emerging mathematical knowledge.

The aim of this assessment approach was to determine the extent to which children recognised the relationship between numerals and quantity as counting and cardinality are one of the first and most fundamental developmental progressions of mathematical learning (Clements & Sarama, 2009). It should however be emphasised that the success of the strategy relied on several key elements: first, the purposeful use of guided questions to engage children's thinking, sustain their attention, and elicit evidence of their mathematical understandings.

Second, the provisions of wooden numerals provided an unequivocal focus on number. Third, the open-ended nature of the activity enabled children to articulate their individual understandings of quantity rather than requiring them to provide the 'right' answer. In this way, the assessment strategy aligns seamlessly with a play-based curriculum and could be implemented by early childhood educators, meeting the requirements of the National Quality Standard (ACECQA, 2011) and identifying children's progression along the mathematics-learning continuum (Clements & Sarama, 2009).

Assessment that is play based supports the many ways in which children spontaneously flow between their ideas as they emerge, evolve and unfold (Wright, 2012). Children develop confidence in their own learning abilities when they are able to connect their new learning tasks to existing knowledge (Bobis, Mulligan, Lowrie & Taplin, 2009). In the same way that educators strive to provide authentic learning experiences, so too can assessment tools be constructed from the child's existing and emergent mathematical knowledge.

References

Australian Children's Education and Care Quality Authority (ACECQA). (2011). Guide to the National Quality Standard for Australia. Retrieved 13 August, 2013, from http://files.acecqa. gov.au/files/National Quality Framework Resources Kit/3-Guide to the National Quality Standard FINAL-3.pdf.

Barden, L. M. (1995). Effective questioning and the ever-elusive higher-order question. American Biology Teacher, 57(7), 423-426.

Baroody, A. J. (1989). Manipulatives don't come with guarantees. Arithmetic Teacher, 37(2), 4-5.

Bobis, J. (2008). Early spatial thinking and the development of number sense. Australian Primary Mathematics Classroom, 13(3), 4-9.

Bobis, J., Mulligan, J., Lowrie, T, & Taplin, M. (2009). Mathematics for children: Challenging children to think mathematically. Frenchs Forest, NSW: Pearson Education Australia.

Carruthers, E. & Worthington, M. (2006). Children's mathematics - making marks, making meaning. London: Sage Publications.

Chase, S. E. (2005). Narrative inquiry: Multiple lenses, approaches, voices. In N. K. Denzin & Y S. Lincoln (Eds.), The handbook of qualitative research (3rd edn, pp. 651-679). Thousand Oaks, CA: Sage Publications.

Clandinin, D. J. (2007). (Ed.) Handbook of narrative inquiry: Mapping a methodology. Thousand Oaks, CA: Sage.

Clements, D., & Sarama, J. (2009). Early childhood mathematics education research: Learning trajectories for young children. USA: Routledge.

Cohen, L., Manion, L., & Morrison, K. (2010). Strategies for data collection: Research methods in education. Abingdon: Routledge.

Cohrssen, C., Church, A., Ishimine, K., & Tayler, C. (2013). Playing with Maths: Facilitating the learning in play-based learning. Australasian Journal of Early Childhood, 38(1), 95-100.

Creswell, J. W. (2008). Narrative research designs. In Educational research: Planning, conducting and evaluating quantitative and qualitative research (3rd edn, pp. 511-550). Upper Saddle River, NJ: Pearson Education, Inc.

Department of Education and Early Childhood Development (DEECD). (2009). Victorian Early Years Learning and Development Framework. Melbourne, Australia: Early Childhood Strategy Division DEECD.

Department of Education, Employment and Workplace Relations (DEEWR). (2009). Belonging, Being and Becoming: The Early Years Learning Framework for Australia. Retrieved 15 August, 2013, from http://foi.deewr.gov.au/system/files/doc/other/belonging_being_and_ becoming_the_early_years_learning_framework_for_australia.pdf.

Doig, B. (2005). Developing formal mathematical assessment for 4 to 8-year-olds. Mathematics Education Research Journal, 16(3), 100-119.

Edwards, C., Gandini, L., & Forman, G. (2011). The hundred languages of children--The Reggio Emilia approach in transformation. Santa Barbara: Praeger Publishers.

Fleer, M. (2008). Everyday learning about howthings work. Canberra, ACT Early Childhood Australia.

Gardener, H. (2011). Frames of mind--the theory of multiple intelligences. USA: Basic Books.

Gelman, R., & Gallistel, C. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.

Ginsburg, H. P, & Ertle, B. (2008). Knowing the mathematics in early childhood mathematics: Contemporary perspectives on mathematics in early childhood Education. USA: Information Age Publishing.

Ginsburg, H. P, Jacobs, S. F, & Lopez, L. S. (1998). The teacher's guide to flexible interviewing in the classroom: Learning what children know about math. Boston: Allyn and Bacon.

Hughes, K., Gullo, D., & Kindergarten Interest Forum. (2010). On our minds: Joyful learning and assessment in kindergarten. Young children on our minds, 65(3), 57-59.

Johansson, B. S. (2005). Numeral writing skill and elementary arithmetic mental calculations. Scandinavian Journal of Educational Research, 49(1), 3-25. Uppsala University, Sweden: Carfax Publishing.

Kilpatrick, J. Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Klibanoff, R., Levine, S., Huttenlocher, J., Vasilyeva, M., & Hedges, L. (2006). Preschool children's mathematical knowledge: the effect of teacher 'math talk'. Developmental Psychology, 42(1), 59-69.

Kress, G. (1997). Before writing: Re-thinking the paths to literacy. London: Routledge.

Kramp, M. (2009). Exploring life and experience through narrative inquiry. In K. Bennett de Marrais and S. D. Lapan (Eds.), Foundations of research: Methods of inquiry in education and the social sciences. Mahwah, NJ: Lawrence Erlbaum.

MacDonald, A. (2009). Drawing stories: The power of children's stories to communicate the lived experience of starting school. Australasian Journal of Early Childhood, 34(3), 40-49.

MacDonald, A. (2012).Young children's photographs of measurement in the home. Early Years: An International Journal of Research and Development, 32(1), 71-85.

MacDonald, A. (2013). Using children's representations to investigate meaning-making in mathematics. Australasian Journal of Early Childhood, 38(2), 65-73.

MacDonald, A., & Lowrie, T. (2011). Developing measurement concepts within context: Children's representations of length. Mathematics Education Research Journal, 23(1), 27-42.

National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM). (2002). Position statement: Early childhood mathematics: Promoting good beginnings. Retrieved 13 November, 2013, from www.naeyc.org/ files/naeyc/file/positions/psmath.pdf.

Perry, B., & Dockett, S. (2002). Young children's access to powerful mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education: Directions for the 21st century (81-111). Mahwah, NJ: Lawrence Erlbaum.

Perry, B., Dockett, S., Harley, E., & Hentschke, N. (2006). Linking powerful mathematical ideas and developmental learning outcomes in early childhood mathematics. Retrieved 13 August, 2013, from www.merga.net.au/documents/RP462006.pdf.

Pianta, R., La Paro, K., & Hamre, B. K. (2008). Classroom assessment scoring system (CLASS) manual, pre-K. Baltimore: Paul H. Brookes Publishing.

Saracho, O., & Spodek, B. (Eds.) (2008). Contemporary perspectives on mathematics in early childhood Education. Charlotte, NC: Information Age Publishing.

Siraj-Blatchford, I. (2005). TACTYC Annual Conference. Birth to eight matters! Seeking seamlessness--Continuity? Integration? Creativity? Quality interactions in the early years. Retrieved 1 November, 2013, from www.tactyc.org.uk/pdfs/2005conf_siraj.pdf.

Siraj-Blatchford, I. (2009). Conceptualising progression in the pedagogy of play and sustained shared thinking in early childhood education: A Vygotskian perspective. Educational and Child Psychology, 26(2), 77-89.

Smith, T, & MacDonald, A. (2009). Time for talk: The drawing-telling process. Australian Primary Mathematics Classroom, 14(3), 21-26.

Stake, R. (2010). Advocacy and ethics: Making things work better. In R. Stake (Ed.), Qualitative research: Studying how things work (pp. 200-214). New York & London: The Guildford Press.

Sun Lee, J., & Ginsburg, H. (2009). Early childhood teacher's misconceptions about mathematics education for young children in the United States. Australasian Journal of Early Childhood, 34(4). 37-45.

Tayler, C., Ishimine, K., Cleveland, G., Cloney, D., & Thorpe, K. (2013). The quality of early childhood education and care services in Australia. Australasian Journal of Early Childhood, 38(2), 13-21.

Thomson, S., Rowe, K., Underwood, C., & Peck, R. (2005). Numeracy in the early years: Project good start. Camberwell, Victoria, Australia: Australian Council for Educational Research.

Tishman, S., & Palmer, P (2005). Visible Thinking. Leadership Compass, July 2005. USA: Graduate School of Education, Harvard University.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner & E. Souberman, trans.). Cambridge, M.A: Harvard University Press.

Walsh, R. (2013). Extending young gifted children through higher order questioning techniques. Giftedness in Early Childhood Conference. Melbourne, Australia, 11 October, 2013.

Wright, S. (2011). Understanding creativity in early childhood: Meaning-making and children's drawings. London: Sage.

Wright, S. (2012). Children, meaning-making and the arts (2nd edn). Frenchs Forest, NSW: Pearson.

Rachel Pollitt

Caroline Cohrssen

Amelia Church

Susan Wright

University of Melbourne

Figure 2. Children's use of wooden numerals Strategies Selected a numeral, but used their own number symbols OT represented the number in a different way, n = 8 Traced around the selected numerical n = 43 Traced around the numeral and integrated their knowledge of quantity into the number symbol n = 35 Used the selected numeral as a guide to copy, n = 25 Note: Table made from pie chart.

Printer friendly Cite/link Email Feedback | |

Author: | Pollitt, Rachel; Cohrssen, Caroline; Church, Amelia; Wright, Susan |
---|---|

Publication: | Australasian Journal of Early Childhood |

Article Type: | Report |

Date: | Feb 1, 2015 |

Words: | 6754 |

Previous Article: | Why is group teaching so important to Chinese children's development? |

Next Article: | What makes people sick? The drawing method and children's conceptualisation of health and illness. |

Topics: |