making mathematics learning visible in early

Narrative assessment: making mathematics learning visible in early childhood settings

Glenda Anthony, Claire McLachlan & Rachel Lim Fock Poh

Mathematics Education Research Journal ISSN 1033-2170 Math Ed Res J DOI 10.1007/s13394-015-0142-2

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Author's personal copy Math Ed Res J DOI 10.1007/s13394-015-0142-2 O R I G I N A L A RT I C L E

Narrative assessment: making mathematics learning visible in early childhood settings Glenda Anthony 1 & Claire McLachlan 2 & Rachel Lim Fock Poh 1

Received: 21 October 2014 / Accepted: 13 March 2015 # Mathematics Education Research Group of Australasia, Inc. 2015

Abstract Narratives that capture children’s learning as they go about their day-to-day activities are promoted as a powerful assessment tool within early childhood settings. However, in the New Zealand context, there is increasing concern that learning stories—the preferred form of narrative assessment—currently downplay domain knowledge. In this paper, we draw on data from 13 teacher interviews and samples of 18 children’s learning stories to examine how mathematics is made visible within learning stories. Despite appreciating that mathematics is embedded in a range of everyday activities within the centres, we found that the nature of a particular activity appeared to influence ‘how’ and ‘what’ the teachers chose to document as mathematics learning. Many of the teachers expressed a preference to document and analyse mathematics learning that occurred within explicit mathematics activities rather than within play that involves mathematics. Our concern is that this restricted documentation of mathematical activity could potentially limit opportunities for mathematics learning both in the centre and home settings. Keywords Assessment . Early years . Mathematisation . Mathematical play Assessment practices in educational settings undergo change as a consequence of alternative theoretical positioning, policy and professional development priorities and accountability demands (Fleer and Quiñones 2013). Despite evidence that the discourse Rachel Lim Fock Poh is deceased.

* Glenda Anthony [email protected] Claire McLachlan [email protected] 1

Massey University Institute of Education, Private Bag 11222, Palmerston North 4442, New Zealand

2

Massey University Institute of Education, Private Bag 102904, Auckland 0745, New Zealand

Author's personal copy Anthony et al.

associated with assessment is increasingly ‘framed within a regime of accountability, measurement and readiness’ (Basford and Bath 2014, p. 121), twenty-first century early childhood education assessment practices depict a shift in focus away from a deficit perspective employing check lists of skills to describe ‘competence’ for the next stage of education. Instead, current practices reflect a growing use of narrative and credit modes of assessment (McLachlan et al. 2013a, b). In New Zealand, Mitchell’s (2008) survey of early childhood centres noted a trend towards ‘more qualitative and interpretive methods of documentation that are able to capture the learning within contexts of relationships and environment’ (p. viii). In particular, the use of learning stories (reported to be used by 94 % of centres) was endorsed as an effective way to assess learning dispositions that are deemed central to the sociocultural framing of the early childhood curriculum Te Whāriki (Ministry of Education (MoE) 1996). How then is mathematics made visible in a culture of assessment dominated by attention to learning dispositions? Knowing that there is an expectation that disciplinespecific areas such as mathematics be assessed within the context of close observations of children as they go about their day-to-day activities within the early childhood setting, and knowing that opportunities for early childhood teachers to engage in mathematical interactions are often under-utilised (Hedges and Cullen 2005; Scoffin 2013), we wondered how young children’s mathematics learning is represented within narrative assessment practices. In this article, we review the assessment practices from three early childhood kindergarten settings. Drawing on teacher interviews and sampling of learning stories, the first section of our findings focuses on how teachers believed mathematics should be made visible within learning stories. We explore how individual and collective philosophical approaches to assessment within each of the kindergartens impacted on what mathematics teachers noticed and chose to document. The second section of our findings summarises ‘what’ mathematics was made visible within a sample of learning stories. Together, these findings suggest that teachers want and need more support in ways to document mathematics learning in ways that can inform planning for learning. Before moving to the research, we first overview the expectations for mathematical learning experiences that are promoted in the early years curriculum Te Whāriki (MoE 1996) and review the theoretical framework underpinning current assessment and professional noticing of mathematics learning within early childhood settings.

Mathematics in early years curriculum The New Zealand early childhood curriculum Te Whāriki (MoE 1996) is founded on aspirations for children to: grow up as competent and confident learners and communicators, healthy in mind, body and spirit, secure in their sense of belonging and in the knowledge that they make a valued contribution to society. (p. 9) Defined as a ‘competence’ curriculum, in which children can influence the choice and sequencing (McLachlan et al. 2013a, b), Te Whāriki emphasises children’s

Author's personal copy Narrative assessment: making mathematics learning visible

competencies, dispositions and theory building through active participation within the social world. As such, curriculum is defined broadly as ‘the sum total of the experiences, activities and events, whether direct or indirect, which occur within an environment designed to foster children’s learning and development’ (p. 10). While this holistic approach has the potential to obscure key mathematical learning outcomes, care is taken to ensure that exemplars of mathematical experiences are woven into the goals and learning outcomes (known as knowledge, skills and attitudes) of the Communication and Exploration strands. For example, teachers are advised to provide experiences for children to: & & & & & & & &

Develop understanding that symbols can be ‘read’ by others and that thoughts, experiences and ideas can be represented through words, pictures, print, numbers, sounds, shapes, models and photographs Become familiar with numbers and their uses by exploring and observing the use of numbers in activities that have meaning and purpose for children Develop skills in using the counting system and mathematical symbols and concepts, such as numbers, length, weight, volume, shape and pattern, for meaningful and increasingly complex purposes Expect that numbers can amuse, delight, illuminate and excite Gain experience with some of the technology and resources for mathematics Develop confidence with moving in space and moving to rhythm Develop confidence in setting and solving problems, looking for patterns, classifying things, guessing, using trial and error, thinking logically and making comparisons, asking questions and explaining to other Develop spatial understanding, including an awareness of how two- and threedimensional objects can be fitted together and moved in space and ways in which spatial information can be represented, such as in maps, diagrams, photographs and drawings. (MoE 1996, pp. 74–88)

Experiencing mathematics is also implicit in the other strands (e.g., developing skills in food preparation (wellbeing); understanding routines and discussing and negotiating fairness (belonging); and discussing/explaining ideas (contribution)). Communication has long been recognised as one of the crucial facets in developing young children’s understanding, reasoning and problem solving skills in mathematics (Jung and Reifel 2011; van Oers 2013; Walshaw and Anthony 2008). In addition to learning concepts and procedures, these experiences also signal expectations for developing dispositions towards mathematics as expressed in the ways they approach mathematical play or activities—be it with delight, confidence, willingness to explore, perseverance and or interest. Carr and Lee (2012) contend that ‘dispositions act as an affective and cultural filter for the development of increasing complex knowledge and skills’ (p. 15). In arguing for the importance of documenting the affective state of the environment that children find themselves in during the assessment moments, Fleer and Quiñones (2013) contend that the nature of these emotional connections to learning concepts can be related to how a learner may ‘behave towards school science [mathematics] later in secondary school or even when studying early childhood teacher education later in life’ (p. 242).


Assessment in early childhood settings Aligned with socio-historical cultural views concerning the provision of opportunities to learn and develop mathematical concepts, Kei Tua o te Pae. Assessment for Learning: Early Childhood Exemplars (MoE 2009) advocates an assessment for learning position. Carr and Lee (2012) regard such learning as situated in the ‘middle’, in the reciprocal relationship between an educational environment and the learning individual. As such, assessment requires that teachers be aware about what—inclusive of social and cultural practices—they are teaching children through play and the role of their mediated interactions in children’s play (Anthony and Walshaw 2009; Hedges and Cullen 2012; Siraj-Blatchford and Sylva 2004). Importantly, deliberate interactions provide opportunities to link everyday and scientific concepts (e.g. wiping a table to the edge and the concept of bounded spaces) that support mathematical concept formation (e.g. area measurement) (Fleer and Raban 2007). In the field of mathematics education, many researchers (e.g. Brodie 2011; Carpenter et al. 1989; Wager 2014) position instruction that embraces assessment for learning as responsive teaching—teaching that involves noticing, attending to and responding to learners’ ideas. These mediating teacher actions of noticing, recognising and responding to children’s learning (Drummond 1993) can be viewed as progressive filters whereby teachers Bnotice a great deal as they work with children, and they recognise some of what they notice as ‘learning’. They will respond to a selection of what they recognise^ (MoE 2004, Book 1, p. 6). The difference between noticing and recognising lies in the application of professional expertise to recognise the significance of what is noticed in relation to the learning and wellbeing of the child. Importantly, recognising requires that educators have a sound understanding of mathematics, are open to the potential for mathematics in many situations and can recognise children’s mathematical competence and strengths (Dockett and Goff 2013). This ability to recognise learning is linked, in turn, to the ability to make a professional response to that learning, and in doing so, to enhance the child’s opportunity to learn either through reinforcement or extension. These mediating actions by an adult are also central to a dynamic view of assessment promoted by early childhood researchers Fleer and Quiñones (2013): Underpinning a dynamic view of assessment … is the idea of adult mediation, which has allowed the assessor to move beyond a static and individual construction of the assessment context. That is, rather than measuring what a child can do on her/his own, dynamic assessment seek to assess the child and the adult working together at a higher cognitive level, where the extent of the mediation is measured alongside what is achieved. (p. 238) So how then should, or could, such professional noticing be documented and for what purpose? Early-years researchers (e.g. Fleer and Quiñones 2013; McLachlan et al. 2013a, b) argue that narrative assessments are well-suited to the ethical, dynamic, forward-oriented and child-focused assessment principles represented in early childhood curriculum. In New Zealand, narrative assessment practices in the format of the learning story framework developed by Carr (2001) constitute the mainstay in early


childhood settings. Within the context of mathematics education, learning stories have also been more recently promoted in the Australian context (Perry et al. 2007). Closely linked to Te Whāriki (MoE 1996), the learning story framework suggests a documented account of a child’s learning event structured around five key behaviours: taking an interest, being involved, persisting with difficulty, expressing a point of view or feeling and taking responsibility. Typically, learning stories are presented as a one page narrative written by the teacher, but may include input from parents and children. Focused on ‘learning in action’ (Carr 2001, p. 49) rather than the individual alone, the expectation is that the narrative explains, rather than displays learning (Forman and Fyfe 1998). In addition to making aspects of children’s learning visible, the narrative is expected to provide explanation of the significance of the event for teaching and learning and propose ideas about ‘where to next’ for the child. Learning stories often include photographs as a record of an aspect or incident of a child’s or group of children’s learning. As with any assessment practice, learning stories can serve multiple purposes. More than just describing learning, they can facilitate discussions about the child’s learning with other staff and parents, document learning over time and support planning decisions about ‘where to next’ for the child. However, there is a growing level of critique as to whether the learning story framework, as it is currently being implemented by teachers, is sufficiently robust for capturing evidence of concept formation or being utilised effectively in planning future learning opportunities (Education Review Office 2007). In particular, researchers have expressed concerns that learning stories downplay domain knowledge. For example, Nuttall (2005) contends that many of the exemplars presented in Kei Tua o te Pae resource (MoE 2009) lack appropriate interpretations of children’s engagement in sophisticated literacy practices, preferring instead to emphasise dispositions such as collaboration and exploration. Blaiklock (2008, 2010) and Fleer and Quiñones (2013) express concerns about the propensity of early childhood educators to interact and assess in the ‘here and now’. Blaiklock (2010) bemoans the frequency of subjective evaluations based on short observations, problems with defining and assessing learning dispositions and difficulties in using learning stories to show changes in children’s learning over time. Similarly, Fleer and Quiñones (2013) express concerns about the focus on ‘looking back on what has been achieved or what is currently being enacted, rather than examining how interaction sequences can orient learning in particular ways to work and think scientifically in the future’ (p. 239). As noted by McLachlan et al. (2013a, b), the corroboration of anecdotal, school reviews and research evidence concerning narrative assessments has fuelled an uneasy suspicion that: too often learning stories focus on children’s disposition and do not capture when children have gained new concepts or understandings. There are few examples in which the learning that is happening is clearly identified….teachers need to avoid just taking pictures and writing descriptive stories and instead focus on using their observations to documents learning that is happening and to inform their curricula decision making. (p. 110) It is with these concerns in mind that we look at how mathematics is made visible within learning stories. In looking at a sample of learning stories that foreground


mathematical activities, our research question is focused on how early childhood educators document and analyse a selection children’s mathematical learning and dispositions as part of their narrative assessment practice.

Research overview The presented findings are drawn from a larger study1 examining assessment practices involving children’s mathematical learning within early childhood settings. The study involved case studies of three kindergartens—Manuka (M), Kauri (K) Rimu (R)— representing high, middle and low socio-economic parent communities, respectively, all located within a mid-size urban setting. In New Zealand, kindergartens provide early childhood education based on the early childhood curriculum Te Whāriki (MoE 1996) for children aged 3–5 years.2Data collection included 13 teacher interviews (inclusive of all teaching staff in each of the centres—coded MT1 to MT4; KT1 to MT4; and RT1 to RT5), purposeful sampling of learning stories from six children’s portfolios within each kindergarten (coded MC1 to MC6, etc.) and interviews with parents of the selected children from each kindergarten. In selecting our sample of learning stories, we wished to explore exemplars of everyday practice regarding the documentation of mathematics learning. To avoid our motives from being construed as auditing of teachers’ work, we invited teacher to direct us to exemplars of ‘good’ practice concerning mathematics documentation. The teachers from each kindergarten were requested to nominate six portfolios based on the following criteria: & &

Portfolios containing learning stories that include mathematics experiences Portfolios of children who have demonstrated ability and interest in learning mathematics

From each of the portfolios, a selection of 3 to 5 learning stories (66 in total) were sampled based on criteria of high visibility of mathematics and sequencing across time. In focusing our attention on those learning stories, we hoped to gain access to ‘what’ mathematics was made visible and regarded by the teacher as important to document for the learner and parent audience. Content analysis of the exemplars was conducted using two lenses: (1) mathematical content and (2) dispositions linked to mathematics activity. All of the teachers participated in semi-structured interviews. Conducted by the one researcher at a mutually agreed location, the interview focused on each teacher’s current assessment practices and perceptions about mathematics learning and assessment in early childhood settings. Lead questions generating the data analysed in this paper included: 1

The larger study was Rachel Lim Fock Poh’s doctoral work. The data collection for this study was completed along with some preliminary analysis before Rachel’s untimely death. Rachel’s supervisors are honoured to be able to posthumously present her findings. 2 In New Zealand, kindergartens provide early childhood education based on the early childhood curriculum Te Whāriki (MoE 1996) for children aged 3–5 years.


& & & & & & & & & &

Can you talk about your current assessment practices? How do you use assessment to inform your programme planning and planning for individual learning? How do you make decisions on what to include and not include within learning stories (for mathematics)? Could you talk about the ways you assess and document children’s mathematical learning? What aspects of mathematical learning do you consider important to document? How often would you include mathematics in learning stories? Are some aspects of mathematics learning more easy or difficult to document? Where do you typically see mathematics learning experiences taking place? How do you analyse mathematical learning in learning stories? Are you aware of parents’ preferences for how mathematics should be documented?

Towards the end of each interview, each teacher was also invited to comment on three of the teacher nominated learning stories (randomly selected by the researcher) in more detail. Copies of the learning stories were used to prompt discussion about how the learning stories came about and perceived mathematical focus, as well as discussion about possible parental follow up. Data analysis of teacher interview transcripts was conducted in two phases. The first phase involved reading all of the teacher interviews using inductive analytic techniques involving comparative within case and horizontal across case analysis (Miles and Huberman 1994). In our interpretative reading of the transcripts, we developed codes associated with common patterns as well as difference that related to assessment practices (e.g. elements of learning stories, purpose, explicit versus implicit content, informal versus formal assessment, etc.). The second phase entailed categorising the codes into emerging themes. Of relevance to this paper are three core themes related to teachers’ assessment practices that emerged from the data: teachers’ philosophical lens, use of a mathematical lens and the role of a collaborative lens.

Learning stories In recounting important elements of a learning story, teachers commented that a learning story should include a combination of child, teacher and parent voice, as well as photos to help the child recall the experience. However, although there was consistency with format across centres, the subtle variations in expectations of the ways mathematics learning was to be documented appeared to be linked to different philosophical positions concerning assessment. While influenced by personal (individual) beliefs and experiences, there was evidence of a signature assessment philosophy negotiated collectively within each centre. For example, the teachers in Manuka kindergarten felt that it was important for them to ‘educate parents about what learning is going on while playing’ (MT1); their role was to interpret the learning that was going on within an activity. As MT3 explains, it was important to make:


learning very visible to parents such as linking learning to research and showing parents what I see children doing and how I relate it to learning. Because as a teacher I can relate playing to learning but it might not be the case for the parents. To make learning visible, their learning stories typically included a description of what happened followed by an analysis that highlighted the valued learning as illustrated in the following extract: No story title Written by MT3 on 03/02/2011 MC3, today you were enjoying grouping various science resources into same family groupings. You and A worked together, and were very precise about where you placed each of the creatures after carefully looking at the colouring, patterning and size of each one. You were delighted to find a large monarch butterfly figurine—excitedly telling me that you have seen some monarch butterflies and caterpillars at your house. MC3, you are confident in using a variety of strategies for exploring and making sense of the world—including looking for patterns, classifying things for a purpose and thinking logically. This work is expanding your maths concepts as you collect, organise, compare and interpret different objects and materials. I wonder where else this fascinating seriation and grouping is occurring at kindergarten?

From a different perspective, Kauri teachers’ learning stories were framed by the belief that teachers should not assume what the child is doing from what they see. As teachers, they questioned their right to interpret, for example, ‘doing a jigsaw puzzle’ as an indication of ‘spatial awareness’. Thus, unless the learning was visible and recognisable as part of a mathematical activity (e.g. one-to-one counting in a numeracy game, sorting in a grouping activity), it was unlikely that the learning story would be interpreted and analysed to highlight specific mathematical learning. This approach is evident in the following two learning stories: the first one avoids any interpretation of mathematical learning, while the second one includes an analysis of explicitly observed mathematics counting skills. No story title Written by KT3 on 09/05/2011 KC4, today I saw you and A spending time at the water trough filling up the different sized water beakers with water. I watched as you worked together to ensure that each one was filled to the top before moving on to the next one. I really liked the way that you took turns at tipping water into each beaker and how you listened really well to your friend A when he called out for you to stop putting water in when the beaker was filled. From just watching you, I felt that you were both just getting enjoyment out of pouring and filling beakers. Perhaps if I see you doing this again it would be a good opportunity to introduce you to some maths concepts such as volume, size and measurement. No story title Written KT4 on 24/5/2011 KC6, I am so impressed, I watched you today on the large mat with the number puzzle and you showed great knowledge as you placed each number in order along with the correct amount of objects. This puzzle was no challenge to you at all, you just whizzed through it so quickly. I can see from other stories KC6 that you have an interest in numbers and mathematics and are often at these types of activities. You showed me that you knew the ordering of numbers from 1 to 10 and could recognise the numerals easily.

From yet another perspective, the Rimu teachers’ focus was to link the child’s learning to the development of dispositions: …looking through a dispositional lens. We talk about the interest and from there it might be dispositions that feed into that interest, e.g., children keep going back to that same thing. [RT3]


Although RT3 mentioned in her interview that her interactions with the child in question involved a lot of mathematical language, the learning story below illustrates that the mathematical learning was frequently back-grounded in preference to the dispositions: U-tonu-tanga/Perseverance Written by RT3 on 14/06/2011 Today RC6, you really enjoyed working your way around the variety of challenges in the outdoor kindergarten environment. It was great to see the way children embraced the challenge of physically working their way over the obstacle circuit. I was intrigued to see the way each child persevered and challenged themselves to manoeuvre up, over and along the course including some great balancing. When I reflect on our curriculum document Te Whāriki, Contribution/Mana Tangata clear links can be made to Essential Learning Areas of Health, Physical Well-Being. Children participating in group physical activities, children develop responsible relationships and respect for cultural perspectives and the contributions of others.

While foregrounding dispositions was a key focus for teachers from Rimu, analysis of all the learning stories reveals that teachers across all of the kindergartens were keen to highlight dispositions (see Table 1), most notably dispositions related to courage, curiosity and confidence. It is also noted that some references were relatively removed from the perceived mathematical activity referring, for example, a reference to responsibility involved ‘assisting [another child] with the puzzle by showing her where the pieces went’. In exploring ‘what’ mathematics was made visible within the learning stories, the mathematical practices/process and content were coded by categories (Table 2) derived from the Te Whāriki curriculum (MoE 1996) and assessment resources (MoE 2009), combined with categories linked to the powerful mathematical ideas proposed by Perry et al. (2007). As way of explanation of the coding, an example of each of the mathematical processes is provided as follows: &

Mathematisation: For young children, mathematisation involves the translating of an everyday situation into mathematical terms often coded in instances of the learning story describing the child using problem solving and/or tools to display thinking. For example, learning stories included descriptions of children’s effort to problem solve the movement of water through a pumping system by coordinating the heights of chambers, and use of measurements to differentiate plant growth. Frequently involving communication—inclusive of verbal and semiotic tools such

Table 1 Learning dispositions referenced in learning stories in each of the kindergartens Learning stories (LS)

Manuka (23)

Kauri (24)

Rimu (19)

Percent of 66 LS

Courage and curiosity

9

10

13

48 %

Trust and playfulness

2

2

1

8%

Perseverance

8

4

4

24 %

Confidence

8

14

13

53 %

Responsibility

1

10

2

20 %

Others (e.g. creativity problem solving, pride)

3

2

4

14 %

Relative frequency per LS

1.34

1.75

1.95


&

& &

as diagrams and gestures—mathematising produces ‘structured objects that allow further elaborations in mathematical terms through problem solving and (collective) reason/argumentation’ (van Oers 2013 p. 188). Argumentation referenced children’s attempts to express reason or justify mathematical ideas, and cases where the teacher/peer introduced mathematical language. For example, a learning story captured a peer’s explanation of how to make a paper chain pattern as ‘If we start with the orange one, then the next colour and then the next, until they are all gone and then start with orange again we will make a pattern of colours, see’ (KC1). Connections referenced explicit links to mathematics activities across different content areas and or contexts. For example, KC1’s learning story notes in to relation to a strong interest in jigsaws that C1 reports completing jigsaws at home. Sequencing referenced a child’s attempts to follow a sequence of instructions. For example, KC4’s learning story described C4’s efforts to construct a Lego plane based on a plan of the pieces needed.

As was the case with references to learning dispositions, the frequency and nature of the references to mathematical practices and content was relatively consistent across the kindergartens. Description of the activities associated with number sense, mental computation, spatial and geometric reasoning and measurement frequently involved building/carpentry with blocks, sandpit play, baking, jigsaws and board games. Of note was the infrequent reference to data or chance experiences. The data set, however, does not tell the full story. Many of descriptions of children’s activities made only scant or passing reference to mathematics or mathematical practices. As noted by MT3, learning stories need to capture significant moments of learning in relation to the child and therefore we take the point that on occasions it may well be valid that mathematics is back-grounded: …for that particular child, it might be that they entered the group of children and they learned to share that equipment. So for them that will be the significant thing Table 2 Mathematics referenced in learning stories in each of the kindergartens Learning stories (LS)

Manuka (23)

Kauri (24)

Rimu (19)

Percent of 66 LS

Mathematisation

14

10

9

50 %

Argumentation

10

12

7

44 %

Connections

3

9

9

32 %

Sequencing

3

6

2

17 %

Number sense

7

13

9

44 %

Algebraic reasoning

9

5

5

29 %

Spatial and geometric

6

3

11

30 %

Measurement

7

5

9

32 %

Data and probability

1

1

0

3%

Relative frequency per LS

2.6

2.7

3.2


that you would write about and you might be talking about their language they are using while they are still doing fantastic number work or they are still grouping or sorting. They are still learning about colours and shapes but your focus might be on the other element. So, it’s knowing those children and seeing what’s significant. However, for many learning stories, evidence of interactive discussions or sustained shared thinking (Siraj-Blatchford 2010) with adults to support mathematisation and argumentation, including the introduction of mathematical language, was limited. For example, KT2 writes ‘I saw you looking at the pump and I think your were trying to work out what was going wrong. I’m not sure how but you must have realised that the water in the lower level wasn’t high enough to move through the pump’. Likewise, RT4 noted in a learning story entitled ‘Numeracy’: ‘I could see that you knew exactly how the game went and had great ideas of sorting the animals …working alongside you today gave me an insight into your wonderful knowledge of numeracy. You have great skills of identifying, classifying and sorting’. Without evidence of discussion, the analysis of the mathematics learning appears vague and unconnected to further opportunities for learning. Moreover, in relation to planning, learning stories often contained very general statements of intent (e.g. ‘We did lots of sorting, matching, comparing and classifying with the fish today. Let’s go fishing again soon’ [MT1]; ‘I’m glad you enjoyed all the predicting we did, especially with the water as we saw such amazing results! Maybe we could do it again another time’. [KT2]).

Discussion In looking to understand why the documentation of mathematics is not visible in ways that reflect a dynamic assessment approach (Fleer and Quiñones 2013), including opportunities for initiative-sharing and collaboration (Carr and Lee 2012), nor appears to motivate significant forward planning (Perry et al. 2007), we revisit analysis of the teacher interviews. Only two teachers reported a strong interest with mathematics with most reporting comfort levels associated with the mathematics of counting and shapes only. For some teachers, their lack of confidence in understanding young children’s potential mathematical development meant that they resorted to descriptions of skills that they observed from a distance. As KT4 noted, ‘watching out for new ways of behaving was evidence of learning’. Moreover, she expressed a concern about her ability to judge whether the child’s activity was ‘truly’ mathematical: I suppose that is a shame that I have not had any training on maths, in terms of ‘is it maths that they are doing or is that saying that I think they are doing maths’? So unless I actually see them doing, like is that because a child is playing with a puzzle, it is not necessary that they are experimenting with shapes and size in the way things fit into the spatial, or is it because they just enjoy doing the puzzles? Despite awareness that mathematics was embedded in many activities (e.g. sandpits, computers, the family corner and games), several teachers acknowledged that they ‘did not include enough mathematics in learning stories’. As KT4 reported, she includes


mathematics ‘say once or twice a week if I am lucky. If I have got a maths lens on it will be more’. RT3 also agreed: ‘If it is really evident in that teachable moment, I think I probably need to get more of a numeracy lens to see it happening’. For most teachers, however, because mathematics was potentially everywhere, there appeared to be a sense that one could be selective about when and what to document. According to KT4, mathematics was more easily visible mostly in the maths area: more inside rather than outside, because the children are quite often sitting down and doing something…I suppose mat time they are doing a lot of mathematical learning because we are actually feeding them with the language and the context of maths. As well as perceived contextual visibility, it appear that for many teachers, the choice of what to document was also influenced by learning that was perceived as easily recognised as part of a developmental trajectory. For example, RT2 commented that she more often picked up ‘those foundational mathematical skills and concepts such as numbers, shapes, colours, grouping and seriation’. Unanimously, teachers argued that working with numbers and grouping and patterning were more easily seen and documented, especially if ‘they are actually working with numbers and using the language as well. So you can definitely say that they are counting out loud or they are talking about shapes or size’ (KT4). Interestingly, several teachers noted that their choice was influenced by what families might be expecting. As MT3 explained: Families often want to know that their child knows things like shape, colour and number and so sometimes you can think, right, they are doing that and so I will write that but they are also doing these other things as well. [MT3] MT4 also noted that that parents’ and primary school teachers’ expectation ‘that children will have a good understanding and recognition of numbers when they start school’ influenced documentation choices. Awareness and expectations concerning school readiness was also noted by KT1 who stressed the importance of ‘counting and awareness of numbers so that they have the basic grounding’ for school. Those activities that were noted as more difficult to document as mathematical included weight and measurement, as well as the more generic spatial activities. Teachers were concerned with the dual purposes of these activities: When a child is doing a puzzle with blocks, is she exploring the mathematical concepts or is she just enjoying sticking the blocks of different sizes together? If she is using the language as well then you can actually document it as mathematical learning, but then I am not too sure if it is so easy to pick up the child’s language. I mean I probably can say you are looking at different sizes and you know that the big blocks had to go on to the bottom before you put the little blocks on. But that is what I am seeing and that is not necessary what the child is doing. (KT4)


When asked to reflect on a particular learning story, they had written several teachers recounted interactions with the child that had not been documented. For example, MT1 noted in reference to a learning story featuring MC3 creating a ladder from mobile pieces that, ‘I talked about the extension ladder, how many pieces there were and how long it was, or if it was taller than Emma and so on’. This contrasted the record in the learning story as: ‘You were happy for me to take a photo of your Fire Engine and demonstrated how long the extension ladder was too’. So while MT1 recounted an example of using a child’s working theories to mediate and extend his understandings, more often the teachers expressed a reluctance to engage in explicit interactions around mathematical talk with the children. MT2, for example, linked this reluctance directly to one’s confidence in mathematics: If you are confident you can ask the questions, you can wonder with the children even if you don’t know the answers but it does help to have the content knowledge, to know where you are heading. A preference to a passive approach to noticing the mathematics meant that for some teachers, their documentation privileged planned mathematical experiences over spontaneous or unanticipated experiences. For example, KT4 noted, despite her awareness, that BI should probably do more than just write this up as a description, I should go back to them [and say] ‘I saw you doing this today, can you tell me what you are doing’?^ her inclination was to structure the analysis of mathematical learning around ‘the things that I can see’. However, without further support for KT4, we predict that this would be unlikely to happen; especially given her belief that children find it difficult to engage in mathematical talk: It can be hard to ask a child what learning you think is taking place for you. Maybe because we don’t use it enough and so they are not used to answering that question. They can’t really sort of verbalise what they are actually learning, they can’t say what they are doing on the monkey bars, you know. Moreover, we see from RT2’s discussion about the process of analysis of mathematical learning why moving away from a descriptive approach could be challenging both mathematically and timewise: If I want to pinpoint what learning is going on and I wanted to word it without me just describing what happened, then I look in the Number Framework book or go back to reading and so I can pinpoint what the child was actually doing, what knowledge they had. In discussions about planning, teachers elaborated that the activities featured in the learning story were frequently targeted activities from previous planning, or that observations of one child’s new skills sometimes prompted them to encourage other children to engage in similar activities. Explicit planning of next steps for individual children as follow up to learning stories was less evident. As MT4 noted:


I kind of get a bit stuck on the whole ‘where to next’ question, which is quite difficult for me because I want it to be something that is really complex and complicated where I am thinking that actually I don’t think it is. It’s just kind of what happens as part of daily actions and events. Overall, we must be concerned that the propensity to document explicit mathematics activities and make pre-planned observations potentially lowers expectations around young children’s ability to engage with complex mathematical ideas (Dockett and Goff 2013).

Implications and conclusions Despite appreciating that mathematics was embedded in a range of everyday activities within the centre, the nature of a particular activity (e.g. whether they perceived the underlying maths to be implicit or explicit—a distinction often related to by accompanying mathematical language) appeared to influence ‘what’ and ‘how’ the teachers chose to document as mathematical learning. Sarama and Clements (2009) suggest that mathematical experiences for young children occur in two forms, ‘play that involves mathematics and playing with mathematics itself’ (p. 327). The teachers in this study, for the most part, felt more comfortable documenting mathematics learning that occurred in the latter context. Where documentation of mathematics learning in the first context occurred, teachers often lacked confidence in their ability to recognise how the children represented their mathematics knowledge and to then build on that understanding through prompting and questioning. Our concern is that this restricted documentation of mathematical learning could mirror restricted opportunities for mathematics learning. As Dockett and Goff (2013) note, ‘if we seek evidence of children’s mathematical understanding in limited ways, we should not be surprised if we find limited understanding’ (p. 772). Given that most of the learning experiences in our kindergartens involve well-planned, free-choice play, it is critical that teachers are able to utilise free-choice play to support mathematics learning. This potential is only realised if the mathematics in play is noticed, explored and talked about, and then documented in a way that enhances planning and communication with parents. Documenting how free-play opportunities can support mathematical learning may be significant in creating educative partnerships with family/whanau. As Sarama and Clements (2009) note, one of the best ways to help low-income children who are the most disadvantaged in mathematics is ‘to help children discuss and think about the mathematics they learn in their play’ (p. 332), and this can be applied to both centre and home settings. The findings of this study, while reflecting only three kindergartens, suggest that teachers would like and need more guidance concerning the assessment and documentation of mathematics learning. Without further tools and targeted professional learning opportunities, teachers like RT2 will continue to undervalue the role of sandpit play—play that if mathematised can provide a valuable opportunity for mathematics learning:


I think what sticks out for me is number knowledge and a child is showing me that she understood numeral recognition or one-to-one counting, doing patterns. I write learning stories about that, but if a child was in the sandpit making a road, using a spade to do that, I probably would write about something else rather than maths even though there is maths going on. In the Australian context, tools, such as the Numeracy Matrix developed by Perry et al. (2012), have proven effective in supporting assessment and planning practices. Our findings suggest that we need further research and support for teachers to make mathematics visible in ways that support early mathematics learning. Our concern is that shortcomings in the capacity to utilise learning stories as an ‘intervention related to co-construction or participation in the learning process’ (Basford and Bath 2014, p. 128) in ways that encourages children and their families/whānau to participate in rich mathematical learning experiences, will potentially open the door to more regulatory and superficial records of children’s play and learning.

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