Kei Tua o te Pae

What’s happening here?

This activity begins with the teacher noticing Quin’s interest in using a ruler and pens. The teacher has ruled some paper into quarters, perhaps to see what Quin will do with them. Quin’s response is to paint a symmetrical pattern.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

The teacher has noticed an interest in rulers and pens, and her response stimulates a development of that interest into an exploration of aspects of geometry: quarters, rectangles, and symmetry. The teacher notes that Quin “appeared to be really happy with” her work and was her own judge of its merits. Quin’s question asks for confirmation: “[T]hey’re quarters, aren’t they?” There is no record here of what came before or what followed, but the documentation provides the opportunity for Quin to revisit the experience and to follow up on this exploration. The teacher analyses the learning with a mathematical lens, noting the “symmetrical look” to Quin’s work and providing what may be a new word to Quin, an addition to her mathematical vocabulary.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

For many children, learning begins by playing with tools with no apparent purpose in mind, and then a purpose or meaning begins to be explored. In this case, the teacher appears to have provided the stimulation with no specific outcome in mind. Quin initiates the direction that her learning takes. The Exploration/Mana Aotūroa strand of Te Whāriki includes outcomes to do with setting and solving problems, looking for patterns, and investigating spatial understandings. Quin explores all these processes in this exemplar.

How does this assessment exemplify developing competence with mathematics?

Both Carlotta (the teacher) and Quin (the child) appear to be exploring aspects of rectangles together: in this case, they are “four quarters”. Quin and her teacher may later explore different ways to represent quarters, or they may explore the horizontal and vertical symmetry of the pattern Quin has created. Quin has very carefully represented the symmetrical pattern: not just in shape but also in colours.

What’s happening here?

This entry in Hamish’s portfolio is part of a series on making corgis, inspired by his pet corgi. Before this point, Hamish has already sewn one corgi and sculptured one in clay. He is very keen to sew another, and a few days before this exemplar is written, the teacher has recorded suggesting that he draw a corgi to be used as a pattern. He does this. The series of “corgi” assessments includes reading, writing, and oral literacy as well as mathematics, and the teacher models the use of these symbol systems and tools for everyday purposes.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

The exemplar documents Hamish’s use of symbol systems and tools for making meaning. His purpose is set in an everyday context, part of the process of getting a task completed. The teacher assists him with his explorations, while the interest and direction come from Hamish. This assessment indicates that the teacher values Hamish’s work: he knows what to do and he remains focused.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

This exemplar illustrates some characteristics of a competent and confident learner and communicator in mathematics. Hamish explores a sustained interest over time, calling on a range of communication media along the way. He comments to the teacher on aspects of the task that interest him. Sewing, including pattern making, is a complex and difficult task, and the teacher comments on two dispositions that are evident here: “His passion for and delight in his ability” and his capacity “to tackle a tricky task – and stick with it”. Hamish’s learning dispositions are being connected to a diversity of purposes and social communities (see page 6): mathematical purposes and the social community of mathematicians.

How does this assessment exemplify developing competence with mathematics?

The purpose of this task is to make a corgi. Hamish is alert to the mathematical ideas in this task, to do with “how many” and “how much”. The teacher picks up Hamish’s mathematical initiatives and includes them in the conversation, helping him to explore them further. To complete this task, Hamish has made a pattern, counts his stitches early in the process, and estimates the number of stitches later in the process. Inspired by the length of wool left over, he measures the width of the table and the easel using a length of wool as a unit, and makes predictions about other comparisons of length (also using the length of wool as a unit). Modelling an everyday purpose of mathematics, the teacher first engages with the task by asking Hamish to estimate how many stitches he has completed with one length of wool (based on Hamish’s initiating comment “I’ve done five stitches”). They then count, as part of the conversation, to see if his estimate is reasonable.

The teacher extends his interest in using a length of wool as a unit of measure. Hamish predicts and compares, using the measure for increasingly complex purposes, developing a “number sense”, exploring the shift from two dimensions (the pattern) to three (the sewn corgi), implicitly posing questions for mathematical exploration (for example, about how much wool is needed for the task), and comparing lengths. These are all purposeful mathematical practices.

What’s happening here?

This is a conversation between the teacher and Nick. It begins with a number question and ends with a discussion about a drawing.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

The teacher is using this opportunity to describe, and provide an example of, some of Nick’s recent development with drawing and of his “inventiveness” (in his thinking about aerials). In her short-term review, she has been very explicit about what she means by “inventiveness”. Nick explains what he is drawing and comments that he doesn’t know why there are two aerials. However, he guesses that it is something to do with the microphones in the helmets. This is not just an interesting story: the teacher links it clearly to learning that is valued. She writes Nick’s words as the text of the story to accompany his drawings.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

The teacher comments that Nick “feels confident to ask for help when he needs it and he really seems to enjoy having an in-depth talk from time to time”. This is a specific commentary on his communication practices. She also writes that he is becoming “more confident and willing to take risks” with writing and drawing, and she notes the way in which this exemplar illustrates his inventiveness with thinking about aerials. Nick is drawing and talking about what he knows from home experiences, a connection to the principle that family and community are an integral part of the early childhood curriculum. The Exploration/ Mana Aotūroa strand includes mathematical processes specified in Te Whāriki, such as classifying for a purpose and making comparisons (ordering the bikes from small to large to emphasise the value of the Harley-Davidson). Nick’s interest in the way in which numbers can denote value is an example of “familiarity with numbers and their uses”, an outcome of the Te Whāriki Communication/Mana Reo strand.¹⁸

How does this assessment exemplify developing competence with mathematics?

This account begins with a question from Nick about the numbers on a price tag. The conversation with the teacher (“What will you spend it on, Nick?”) moves the conversation on to motorcycles, a topic that Nick knows a lot about. When the teacher then suggests he draw a picture about this, he draws four bikes and “ordered them from small to large”. The ordering is in terms of the size (length) of the bikes (and, perhaps, their status in Nick’s eyes): Nick can name the type of each motorbike and he includes his mathematical ideas (about measuring their size and speed) in his account of his drawings.

What’s happening here?

The teacher illustrates the use of a calculator, then temporarily loses it as one of the children takes it over for a purpose of her own.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

Two events are illustrated here, both of them valued enough to be documented. The teacher records Lute’s interest in the calculator, and she notes that Lute “was as if ... oblivious to her surroundings” when using it. Now that this story has been documented, other teachers (and Lute’s family) can notice and recognise opportunities to strengthen Lute’s interest – in both writing numbers and in using the calculator.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

This exemplar documents the participation of the children in an enterprise of great importance: applying for a grant for equipment for their early childhood centre. They belong to and are engaged in a learning community with connections to the world outside the centre (the market – represented by the catalogue – and the funding providers). Te Whāriki emphasises using mathematics in activities that have meaning and purpose for children. It also has the learning outcome “Children develop the expectation that numbers can amuse, delight, illuminate, inform, and excite”.¹⁹

How does this assessment exemplify developing competence with mathematics?

In this assessment, a group of children and their teacher are participating in a real-life mathematical activity. They are working together on the budget for an equipment grant. The children choose what they want from the catalogue, and the teacher uses a calculator to add up the cost of their choices.

Lute writes the names of her family on the blackboard nearby during a halt in the budgeting enterprise. She then borrows the calculator to press keys, watching the numbers come up on the calculator screen, reading them out loud, and writing them down. She is developing competence with the symbols (numbers) of mathematics and with a mathematical tool (the calculator).

What’s happening here?

The teacher suggests measuring a length of play dough when Tom comments that “it’s sooo long!”

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

This documented assessment in Tom’s portfolio records in detail the sequence of his measurements, including photographs that Tom will be able to read himself. Tom can therefore revisit this process. The short-term review even includes a photograph of part of a ruler. The teacher notes the specific mathematics learning and suggests a possible next step – measuring Tom himself.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

This record illustrates responsive and reciprocal relationships between Tom and his teacher. This is a good example of pedagogy that incorporates outcomes from the Communication/Mana Reo strand. The teacher is helping Tom to develop skill in using the counting system and to strengthen his learning about the mathematical symbols and concepts of numbers and length. The initiative is shared. She adds a tool for measurement, adds the word “centimetres” to Tom’s measurement, sets a challenge (30 centimetres), documents the process, and suggests a next step.

How does this assessment exemplify developing competence with mathematics?

This assessment is about using units of measurement as Tom sets himself the task of making piped lengths of play dough longer and longer. The teacher introduces the idea of measuring by numbers of centimetres using a ruler, and Tom readily practises this task. He learns that, by using a ruler, he doesn’t have to compare the lengths of dough against each other. He can remember the length in centimetres of the longest one so far. If the proposed next step is taken (measuring how tall Tom is), it might well be preceded by some estimation (the teacher has already signalled this possibility when she predicts “That would be a great big number”). Tom is learning the use and value of a mathematical tool (a ruler) while learning about a unit of measurement (centimetres). The teacher also records that Tom has used an accurate method of measuring by his making sure that the edge of the dough strip corresponded with the beginning of the ruler. The photographs in the assessment provide a record of this method for Tom to refer to on future occasions.

What’s happening here?

This is a series of stories about Ezra as he experiments with ways to make himself taller, to balance, and to measure height.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

The continuity of Ezra’s explorations is documented as he uses a number of tools (sawhorse, ribbon, bricks, cardboard tube) to change and measure height. The teacher observes carefully, adding suggestions or resources at appropriate times. A voice from home is included in the documentation, connecting Ezra’s measuring experiences at home with those at the centre. The teacher revisits the documentation with Ezra, and he describes his experiences by reading the photographs. He also recognises the letters of his name.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

Connections with home are included here as the family make links with the curriculum at the centre. When the assessment is shared with Ezra’s family, they make a contribution to the record of continuity over time and place. Ezra uses a successful strategy for including others in the play: inviting them to jump with him.

This exemplar is an example of a mathematical pathway that is linked to the Te Whāriki principle of reciprocal and responsive relationships with people, places, and things. The Communication/ Mana Reo strand emphasises mathematics learning through “activities that have meaning and purpose for children”.²⁰ The mathematics in this assessment is sited in events that absorb Ezra and hold meaning for him.

How does this assessment exemplify developing competence with mathematics?

This documented assessment records Ezra’s developing understanding of the concepts of height, measuring height, and number. Ezra experiments with tools that help him to reach a greater height and then with artefacts that help him to measure height. Throughout this process, he demonstrates the activity of “playing” when other children join in the jumping game he has instigated.

Ezra measures using string and a cardboard tube, and he compares his results of different trials (measuring himself and then Eisak against the tube). He uses vocabulary of comparison. “Am I taller than this?” The teacher includes counting in Ezra’s activities by leading him in a backwards countdown of three for jumping and then by encouraging him to count the number of bricks that he needs to create a firm foundation for standing and balancing.

What’s happening here?

This is a series of stories about Jack exploring space.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

A number of these assessment stories focus on Jack exploring space and his place in it. Another illustrates his interest in posting and stacking. These assessments provide Jack’s teachers with ideas of “what next?” for Jack.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

Jack crawls, climbs, and stacks and posts objects. His explorations with these resources, along with his exploratory play with other children, are recorded in these assessments. They indicate the range of resources Jack has available to enable him to explore spatial relationships.

The Exploration/Mana Aotūroa strand of Te Whāriki includes learning outcomes that support: children’s confidence in moving in space, their increasing control over their own bodies, and their manipulative skills, agility, co-ordination, and balance.

How does this assessment exemplify developing competence with mathematics?

Jack is exploring ways of using space, especially in relation to his own body. Through such exploration, he is developing his perception of depth and distance. He is also exploring the shapes and properties of materials and play equipment. The New Zealand Curriculum describes geometry as involving “recognising and using the properties and symmetries of shapes” ²¹ and includes “Sort objects by their appearance” as a level 1 achievement objective. Within this strand of Geometry and Measurement, geometry also involves “describing position and movement." ²²

What’s happening here?

This assessment is one of several items in a series on doing surveys at the early childhood centre. Children carried out surveys on the colour of their socks and tops, on pets at home, on where they love to play, and on their bags. The teachers helped them to convert the data into graphs. In this exemplar, Jake surveys the children’s bags hung up over their lockers in the cloakroom. 

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

The surveys were displayed on the wall so that the children could compare different ways of displaying survey data. Individuals’ surveys, like Jake’s, were also included in the children’s folders so that they could revisit them. The teacher responds to Jake’s initiative to do a survey by prompting him to refer to previous work when he gets stuck. She provides advice on request and encourages discussion among the children when Jake can’t discover a solution to his problem. These strategies for “finding out” have been documented and can serve as reminders for the children about what to do when they’re “stuck”. The teacher checks Jake’s system for recording the bags and verifies that his method will provide accurate data.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

In Te Whāriki, being a learner – and in this case being a learner of mathematics – includes calling on the resources of people, places, and things to assist with making sense of the world. This is exemplified here. Jake consults the teacher, and one of the aspects of difficulty (multicoloured recording) also becomes a topic for discussion by the children. Jake also uses the wall display (things in the environment that help with learning) and is participating in a place where surveys are of interest and are carried out in a range of ways. Jake’s engagement in the process of doing surveys illustrates the learning outcome in Te Whāriki that children will develop the expectation that numbers can “amuse, delight, illuminate, inform, and excite”.²³

How does this assessment exemplify developing competence with mathematics?

Jake’s purpose in this learning story is to gather information about his environment, and he uses a mathematical tool (surveying) to do this. The New Zealand Curriculum states: “Statistics involves identifying problems that can be explored by the use of appropriate data, designing investigations, collecting data, exploring and using patterns and relationships in data, solving problems, and communicating findings.” ²⁴ While Jake is mainly collecting data in this assessment, he is also considering how to record data (on multicoloured bags) appropriately. Also, his choice of a recording method that ensures that the survey will include all the bags shows that he has thought about the design of his investigation. The teacher records that Jake “likes to discriminate, sort, match, count and record”.

What’s happening here?

Trujon is building a garage with a group of children. Photographs of the work in progress and of the completed work were taken and annotated. A learning story about this activity is also documented.

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

This assessment in Trujon’s folder reflects the learning community (his teachers and his family) noticing, recognising, and responding to Trujon’s learning. The photographs are annotated in Sāmoan, and the learning story and parent comment are also in Trujon’s home language. The comments beside the photographs may have been dictated by Trujon. The note beside the third photograph says: “The garage is completed. The cars are brought in to sleep. That’s the front entrance where the cars come in.” The teacher writes an evaluation of Trujon’s progress at the centre, commenting on him working with other children and his interest in carpentry. An invited parent comment describes what Trujon enjoys doing at home, noting that his interests include carpentry. This is a collaborative record.

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

This learning story illustrates Trujon learning from experience how to design a stable construction. Trujon learns from mistakes (the building falls down a number of times). He remains involved and focused on a task. (He works for one and a half hours on this building.) Trujon also displays leadership skills, managing the group with instructions like “Hold onto this”, “Don’t let it fall”, and “Put this on top.” This exemplar shows how children supporting each other and learning together, a key feature of play, makes a valuable contribution to mathematics learning.

How does this assessment exemplify developing competence with mathematics?

In this assessment, Trujon undertakes the collaborative and purposeful task of building a garage for the toy cars. He leads the building process by giving instructions on how to create a solid structure from unit blocks. The different lengths of the blocks are utilised to make a structure that doesn’t fall down. This is accomplished by using long blocks down the sides and half blocks across the top. The decorative design is created with triangles placed on small rectangles. Working with unit blocks gives Trujon the opportunity to gain a working knowledge of geometric shapes. The New Zealand Curriculum highlights “using the properties and symmetries of shapes”.²⁵ Unit blocks provide just such an opportunity. A level 1 achievement objective for position and orientation is “Give and follow instructions for movement that involve distances, directions, and half or quarter turns".

What’s happening here?

This exemplar summarises some of the work in Jessica’s folder as she experiments with patterns, inspired by a visit to the Māori Gallery at Auckland Museum. During this visit, the children observed and drew the patterns, including the kōwhaiwhai. Not all Jessica’s artwork is included here, only the drawing that came from home and some of the koru pattern drawings. 

What aspects of noticing, recognising, and responding to mathematics learning does this assessment exemplify?

This drawing was done at home, but her mother recognised its connection with the artwork of patterns and people that Jessica had been exploring at the early childhood centre. The teacher adds the home drawing to Jessica’s folder, and she comments on the development of her artwork over time. She also acknowledges the uncertainty of her analysis: “One can only wonder if she is experimenting further with repeating a pattern ...” Jessica’s voice is here too. She explains that she has drawn stands on the photograph frames. “Look, these make them stand up ...”

What does this assessment tell us about mathematics learning (using a Te Whāriki lens)?

In Te Whāriki, outcomes are summarised as working theories and learning dispositions. In terms of learning dispositions, the teacher comments that Jessica is “observant” and “creative”. Jessica is also exploring a working theory about patterning, in a range of contexts.

A principle in Te Whāriki is that the wider world of family and community is an integral part of the early childhood curriculum. There is a family contribution to Jessica’s work, and her exploration of pattern has been inspired by a visit to the local museum. Her mathematics understanding is becoming connected to a greater diversity of purposes, places, and social communities (see page 5).

How does this assessment exemplify developing competence with mathematics?

Jessica has been playing with connected patterns, and her drawing is one example of this interest. The teacher points out the mathematics in her analysis of the drawing: she records that she and Jessica have discussed koru patterns and implies that Jessica has been motivated by her observations and these discussions; she sees connections with Jessica’s other work on repeated patterns. The New Zealand Curriculum states that algebra “involves generalising and representing the patterns and relationships found in numbers, shapes, and measures”.²⁶ Jessica’s repeated and sequential pattern displays a “rule” or a relationship: as the figures get smaller, the legs get longer. Such “rules” are part of explaining, one of the six activities identified by Bishop that are responsible for the development of mathematical ideas in any culture.²⁷