2.3 Thinking about demonstrating literacy and mathematics behaviours

For a wide range of practical examples of how children and educators interact to make thinking and learning about literacy and mathematics visible, in connection with related overall and specific expectations in the Kindergarten program, see the expectation charts for this frame in Chapter 4.5.

See also the section “The Role of the Arts in Kindergarten” in Chapter 3.1 for important information related to this frame.

Literacy behaviours: What are we learning from research?

In the knowledge economy, memorization of facts and procedures is not enough for success. Educated workers need a conceptual understanding of complex concepts, and the ability to work with them creatively to generate new ideas, new theories, new products, and new knowledge. They need to be able critically to evaluate what they read, be able to express themselves clearly both verbally and in writing, and understand scientific and mathematical thinking. They need to learn integrated and usable knowledge, rather than the sets of compartmentalised and de-contextualised facts. They need to be able to take responsibility for their own continuing, life-long learning. (OECD/CERI, 2008, p. 1)

“Literacies” is a broad term used to describe the development of the physical, emotional, social, creative, linguistic and intellectual means of communication among young children. (British Columbia Ministry of Education, 2009, p. 29)

We know more about the capabilities of young children than ever before. There is much evidence demonstrating that children are becoming literate in a wide variety of ways (Wien, 2005; Luke, 2007; Dickinson & Neuman, 2005; Sulzby & Teale, 1991). Thinking about literacy in the broadest possible way is therefore critical to helping children develop their ability to understand and communicate – for example, the ability to understand verbal and non-verbal aspects of communication (including emotional, social, and physical cues); to think critically about what they see, hear, and read; and to express themselves by using language in a variety of creative ventures. The development of this broad literacy in young children provides them with a strong basis for successful learning throughout their lives.

Literacy behaviours are evident in virtually everything we do, say, and represent. According to literacy scholar and educator Allan Luke, the challenge for today's students is that they are being asked to read not just the text, but the world. Children need to [shape] and [master] a repertoire of capabilities (Luke & Freebody, 1999, p. 2) to enable them to move beyond basic comprehension skills to understanding and using texts on several levels for a range of purposes in a range of technologies. To help meet this challenge, it is important for educators to give children many opportunities to use and develop literacy behaviours – for example, to use language to describe, to give reasons, to ask questions, or to negotiate – in a wide variety of contexts. It is equally important for educators to learn to use many “languages” both expressively and receptively, to broaden their understanding of what it means to be “literate”.

Children's prior engagement with literacy outside the school

The foundations of language development, literacy, and the capacity for relationship begin to be established at birth. Infants just a few hours old communicate by using facial expressions and responding to adult facial expressions and by using cries and other sounds that vary with context and need. Children continue to develop their ability to communicate, both non-verbally and verbally, through interaction with adults and children at home, in child care, and in the community before they go to school, and they do so in a wide variety of ways.

Children beginning school may demonstrate a prior knowledge of literacy from their daily experiences by, for example:

• asking questions and expressing their thoughts, feelings, and opinions;
• retelling or dramatizing familiar events or stories;
• saying “I knew it said ‘spider” because I used the picture”, or “It's a ‘T”, because it starts just like my name”;
• noticing and naming non-verbal communications (e.g., “I saw by her face that she was angry”).

In addition, before they begin Kindergarten many children may also be able to:

• talk about stories that they have read with a family member (e.g., provide predictions in answer to a parent's questions, such as “Where do you think they might be going on the bus?” or “What do you think they will do when they get to the park?”);
• read a wide variety of materials such as environmental print (e.g., traffic signs, stop signs, street signs, logos, signs in stores and restaurants and on the subway or bus), children's books and magazines, instructions, and labels on boxes and cans in the grocery store;
• write for a variety of purposes (e.g., shopping lists, notes to others, labels, recipes, instructions);
• use non-verbal communication to share thoughts, feelings, and emotions.

In any case, it is essential for Kindergarten programs to build on the knowledge and experiences that children already have when they come to school. It is also essential to keep in mind that children come to school with vastly different experiences and kinds of exposure to literacy. All young children need learning experiences that help them understand the world around them and enable them to develop their ability to communicate. It is therefore important for children to have rich and engaging learning experiences that are relevant to their lives and that provide opportunities for them to develop their capacity to listen, observe, and think and their ability to express themselves.

To help maintain reciprocal relationships between home and school, educators can encourage parents and other family members to continue to engage their children in literacy opportunities at home, and to share those experiences with the educators. Educators can communicate with children's families in various ways (for example, through telephone calls or e-mail, or through translators) about the importance of support from adults or siblings who listen and respond to what young children say, who read to them frequently, who have discussions with them, and who model reading and writing in any language. Listening to someone reading stories and other kinds of texts enables children to learn new words; to become familiar with the patterns, rhythms, and structures of a language; and to extend their experiences.

Supporting the development of literacy behaviours

Literacy is essential to enable a child to succeed in school and in later life. The educators should become familiar with the stages in the process of learning to read and write, and should use this knowledge when thinking about possible literacy experiences for young children and when observing and assessing their learning to help them continually acquire literacy skills.

All children can benefit from classroom experiences that focus on literacy development. To maintain high expectations for all children, it is important for the educators to build on a child's strengths and focus on what the child is already able to do – that is, to see all children through an “asset lens”. It is also important for the team to make adjustments to learning opportunities on the basis of ongoing observation, conversations, documentation, and analysis of their observations in order to maintain a zone of proximal development for the child.

Literacy behaviours are evident in virtually every aspect of human behaviour. Young children may demonstrate literacy behaviours by, for example:

• sharing their ideas, feelings, interests, and experiences;
• looking at name tags to figure out who class visitors may be;
• asking and responding to questions;
• noticing letters and words no matter where they appear;
• showing an interest in print;
• initiating, responding to, and engaging in both verbal and non-verbal communication with others.

In the earliest stages of literacy development, children:

• ask simple “what” and “where” questions;
• mimic the reading process;
• begin to understand what reading is and how it works;
• learn that what they say can be written down;
• use simple vocabulary to describe things;
• engage in pretend play that includes language;
• listen to stories.

As they assimilate this understanding, children:

• learn to pay attention to the way print and books work;
• learn that printed letters and words represent the sounds and words of oral language;
• listen to each other with attention and engage in give-and-take conversations (turn taking);
• become aware that some words rhyme or start or end in the same way, and thus begin to develop phonological awareness;
• begin to share their ideas and responses to texts in a variety of ways;
• learn that writing can communicate a message;
• begin to explore different purposes for writing;
• represent their thinking graphically by drawing, painting, dramatizing, sculpting, building, and gesturing;
• express their thoughts and ideas with increasingly extensive and specialized vocabulary;
• ask and respond to questions that demonstrate and require predicting, making inferences, connecting, and critiquing.

In their written representations children demonstrate literacy behaviours by, for example:

• including pictures and symbols;
• matching spoken words to written words;
• using familiar or high-frequency words;
• using approximate spellings of words, based on their ability to hear, identify, and manipulate sounds (phonological and phonemic awareness) and on their knowledge of letter-sound correspondence (phonics);
• leaving spaces between words, thus showing an understanding of the concept of words;
• revealing a developing sense of “voice”.

When thinking about possible learning experiences, educators consider children's cognitive, communication/language, physical, social, and emotional development. The most successful learning takes place when the educators provide literacy experiences that are based on an understanding of the child's total development. For example, the child may become frustrated and discouraged if the literacy learning is beyond his or her cognitive ability or if the demands of the learning call for greater social or emotional maturity. It may be necessary to scaffold some aspects of the learning experience to make it suitable for particular children. The educators therefore need to document the strategies used for communication by and among children, determine a learning opportunity that is appropriate for the child, and decide when and how to intervene to make thinking and learning visible to both the child and the educator.

Children need to experience language and literacy concepts in depth through repeated investigations over a long period of time. Such concepts include the following:

• Letters are used in different ways (e.g., to make words; to create patterns; to label things; to identify something in combination with numbers, such as an apartment number [Apt. 2A]).
• Letters that appear in personal names appear also in other words. They may be pronounced in the same way or in a different way.
• Vocabulary used in conversation may differ from vocabulary used in books.
• Readers use the words, the illustrations, and their prior experience to make sense of text.
• Readers use different strategies (e.g., look at the picture, look for a little word in a bigger word, use letter-sound recognition, think about the context) to decode words.
• Some words that are used frequently in speaking, reading, and writing are words that everyone has to know from memory.
• What a person thinks and says can be written down for other people to read.
• Writers use different sources to support their writing (e.g., word walls; environmental print; help from a friend; their own knowledge of letters, sounds, and words they have in their own memory).
• Speaking and listening involve respect and reciprocity in order to be satisfactory for each person involved.

It is important that learning experiences in language and literacy allow children to see themselves as individuals who talk, listen, read, write, and view media texts, whether the experiences are intended for small groups or individual children or for use in the classroom or the outdoors. Children need to understand that all forms of communication help them to make sense of their world.

Understanding the importance of first languages

The language worlds of young immigrant children are rich and varied. They do not enter the classroom as blank language slates. … [T]hey arrive in the classroom as active language learners and users. (Chumak-Horbatsch, 2012, pp. 3, 24)

It is important for educators and children's families to work together to support the continued development of a child's first language. Educators can encourage families to continue to use their own language at home as a foundation for language and literacy development in English – for example, family members can tell or read stories to their children in their own language. It is also important to find opportunities to bring children's first languages, including sign language, into the classroom – for example, by reading dual-language books in class or inviting family members or other speakers of the language in the community to act as resources. Knowing about one's heritage and culture reinforces not only the value of maintaining the first language but also the development of a positive cultural identity and an increased sense of self-esteem and security.

The following are some additional reasons for children to continue to develop proficiency in their first language:

• Continued use of the first language allows children to develop age-appropriate vocabulary related to their knowledge of the world around them without having to wait until they have learned enough English to engage with such topics.
• A rich store of knowledge gained in the first language will transfer readily into the second; for example, it is much easier for children to learn vocabulary connected with the ways in which certain objects “match” if they already know the vocabulary in their first language.
• Reading and storytelling in the first language – including languages with non-alphabetic writing systems – strengthen children's understanding of fundamental and universal literacy processes. For example, children learn that thoughts and ideas can be written down, and that when they are written down they can be read by others. When listening to storytellers, children absorb the structure of the language, and this knowledge can then help them to make sense of unfamiliar written texts.
• Children who see their previously developed skills acknowledged in school are more likely to feel confident and take the risks involved in learning in their new environment.
• Children can see English as an addition to their first language, rather than as a substitute for it.
• Children who know another language have already learned the important lesson that words are not the things or actions themselves but represent those things or actions. Knowing this results in mental flexibility and makes it easier for children to acquire further languages.
• All children who continue to develop a strong foundation in their first language as they learn other languages are well prepared for participating in a global society.

Using critical literacy to develop children's critical thinking

Preparing young children to be literate in their fast-paced, technological and multiple text world requires educators to reflect upon and challenge their own beliefs of literacy. The learning of functional literacy skills is important, but it cannot overshadow the opportunities presented from incorporating critical literacy pedagogy. (Harwood, 2008, p. 9)

In recent years, critical literacy has increasingly been seen as a skill that will enable children to navigate a text- and media-saturated world in order to meet the challenges of an ever-more-complex society. Critical literacy involves looking beyond the literal meaning of a text in order to analyse and evaluate the text's complete meaning and the author's intent. It is considered an essential skill based on the awareness that language, including literacy, is a key means through which we construct, understand, and express our world view – that is, our view of ourselves and of others (Luke & Freebody, 1997). It cannot, therefore, be seen as a piece of knowledge but must become a central approach, or a stance or culture of thinking (Hadjioannou & Fu, 2007, p. 47), in early years classrooms.

Proponents of critical literacy embrace the notion that children need both basic literacy and critical literacy to help them come to terms with the many forms and types of text that surround them. For this to happen, children must see their classroom as a place where they can safely ask questions, examine their own and others' viewpoints, clarify their thinking, and take a stand on the issues and relationships that are important to them and their future. A learning environment that is respectful and that is co-created with the children promotes the development of skills such as risk taking and inquiry that are fundamental to critical literacy and critical thinking.

Young children have proven time and time again that they are capable and competent in discussions. They have shown themselves to be willing to participate in conversations about topics that are meaningful to them and that have an impact on their lives. It is sometimes the adults who feel challenged when approaching “difficult” issues with young children, perhaps because they feel uncertain about how to talk about such topics with young children. In a Kindergarten classroom, use of a broad range of “languages” can engage children in exploring and examining issues such as bias, point of view, fairness versus unfairness, and the related equity and social justice concepts that naturally arise, while acknowledging that some issues may be more sensitive for some children than for others

Keeping all these considerations in mind, educators can provide multiple opportunities for children to develop critical literacy skills by:

• providing entry points for discussion of the children's questions and wonderings;
• reading aloud with the children and asking questions to stimulate discussion;
• noticing and naming behaviours in the classroom that can provoke discussion (e.g., “We've noticed that more boys than girls play with the blocks. Why is that? What can we do about it?”).

For example, after reading about a social issue that is important to the children, the educators may ask questions to focus and scaffold discussion, such as, “Someone wrote this story. Who do you think it's written for?”; “Let's look at it from the point of view of J. ...”; “Whose voice is missing?”; “How could the story be told differently?” By engaging children in such discussions, and encouraging them to ask questions, educators provide them with opportunities to question their understanding of issues that arise in the classroom, in a storybook, or among their classmates. Such discussions can take place not only during a read-aloud or shared or guided reading but also in other contexts where similar issues arise, such as in the blocks area or at the sand table.

Literacy learning throughout the day

The development of literacy through literacy behaviours is not limited to a particular time in the day. Literacy learning is incorporated throughout the day – it can be made visible, or explicit, to the children in any context, and can be observed by the educators at any time.

By focusing on literacy behaviours, educators can find many experiences throughout the day that can be used to develop children's literacy. The literacy behaviours of using language to describe, to give reasons, to ask questions, and to negotiate are observable in multiple contexts – for example, negotiating during a block construction, discussing plans at a sand table, describing intentions while painting, asking questions during a conversation, giving reasons for moving an object to another table, modelling behaviour during a read-aloud, or discussing options during interactive writing with a small group.

The educator team members play a critical role in engaging children in literacy behaviours throughout the day by creating a supportive environment (including varied contexts and materials) for using language throughout the learning areas in the classroom. The following are some examples of ways in which the educators can engage the children:

• providing frequent opportunities to listen to poems, songs, and rhymes and to read stories and non-fiction texts together by noticing and naming specific literacy behaviours (e.g., “Isabel, I notice that you put periods at the end of your sentences. That is what good writers do.”)
• asking the children questions that elicit descriptions or explanations of their thinking processes (e.g., “Why do you think that …?”; “What should we do next …?)
• asking questions that help the children make connections between what they already know and what they are seeing, reading, or learning at the moment
• encouraging the children to talk about what they notice or give reasons for what they prefer
• asking questions that encourage discussion and/or negotiation between the children (e.g., discussion of reasons why a block tower fell, negotiation about taking turns reading a picture book)
• inviting children to pose their own questions and/or to investigate the ideas that they are proposing
• modelling beginning reading and writing strategies by “thinking aloud” in all areas of the classroom; for example, at the sand table or in the blocks area, or during read-alouds, shared or guided reading experiences, or interactive writing experiences
• showing the connection between talking and writing in practical examples that arise from inquiries that the children are pursuing (e.g., turning a list of items they are discussing into a written list)
• incorporating literacy materials in all areas of the classroom to help the children see that reading and writing, and talking about them, are all meaningful aspects of their daily lives
• including the children in negotiations about choosing literacy materials that are interesting to them
• giving the children time to explore their experience of language throughout the day rather than at a specific time in the day

In order to provide meaningful literacy experiences for the children, the educator team members also do the following:

• discuss and reflect on the impact of their teaching approaches throughout the day
• observe the children's literacy behaviours, document them, and analyse the documentation
• regularly review the pedagogical documentation (e.g., watch videos to check for progress)
• determine where to go next in the learning on the basis of observations of the children's interests, strengths, and areas for improvement (assessment for learning)

Making decisions on level of support

In order to make thinking and learning more visible to the child and to others, educators have shifted away from considering that “gradual release of responsibility” for learning takes place at a specific time in the day; instead, educators have moved towards thinking about the “level of support” that the child or group of children requires, and then finding an appropriate context in which to offer the support. For example, in the case of literacy learning, the responsibility for learning about an aspect of literacy could be gradually released to children at any time in the day, such as when the children are playing with blocks or engaged in an inquiry about worms. In other words, modelled, shared, guided, or independent learning opportunities are provided according to the level of support that the learner needs, regardless of the context in which the learning is taking place.

The provision of appropriate levels of support does not always unfold in a linear way from modelled through to independent learning contexts – for example, educators may use a shared learning context to give support to children who bring a significant level of prior knowledge to their learning. In the case of literacy learning, the educators can determine the level of support needed by the children by observing what the children already know. The educators can then use a variety of levels of support throughout the day for individual children, small groups, or large groups, and in learning areas inside the classroom or in the outdoors.

Educators make decisions about the provision of appropriate levels of support many times throughout the day. The analysis of the pedagogical documentation is critical to the decision-making process. Educators collaborate with each other and with the children to analyse their observations, and then negotiate the context and timing of learning opportunities with the children. Often, as a result, the same child may receive different levels of support in different contexts.

Decisions on the level of support needed are also based on the level of development of each child. The goal is to offer support within the zone of proximal development for that child – at a level that makes the task neither too challenging nor too easy – and to provide the scaffolding that will enable the child eventually to demonstrate the learning independently.

For more information about pedagogical documentation, see Chapter 1.4: “Assessment and Learning in Kindergarten”.

Educators use their professional judgement when selecting appropriate tools to help in decision making about next steps and level of support for children. Developmental continua and learning trajectories are common observational tools used by educators. For example, to help make decisions about how much support to provide in reading, educators can gain insights into children's reading by using an observational tool that places emphasis on the process, since the process can be as important as the result, particularly for children who have gained some experience in reading (Clay, 2000).

Literacy and the learning environment

Educators and children co-create a responsive literacy learning environment in all learning areas. Together they pose questions, share theories, communicate ideas, and reflect on their observations throughout the day. An effective literacy learning environment in Kindergarten is responsive to the needs, interests, and wonderings of the children, builds on their strengths and preferences, and provides opportunities for them to share their thinking in many different ways. When educators are aware of and able to understand and respond to the many “languages” children use to communicate, they give every child a “voice”.

In Kindergarten, literacy learning thrives in an environment that:

• fosters positive attitudes and beliefs (a “growth mindset”) about learning and literacy development;
• involves co-construction of a safe place for expressing opinions, questioning, taking risks, innovating, and establishing agreed-upon ways of learning together so that every voice is heard;
• ensures that all learners see themselves – their interests, values, cultures, and perspectives – represented in the learning environment;
• provides a variety of ways for children to communicate their thinking and learning;
• includes spaces where children can talk, listen, read, and write;
• makes materials for communicating accessible to the various learning areas.

See Chapter 1.3, “The Learning Environment”.

Mathematics behaviours: What are we learning from research?

High quality instruction in mathematics and high quality free play need not compete for time in the classroom. Engaging in both makes each richer, and children benefit in every way. (Sarama & Clements, 2009a, p. 331)

Research supports the understanding that mathematics experiences occur naturally as children play. During play, young children spontaneously measure, sort, classify, estimate, pattern, count, and more (Ginsberg, 2006; Sarama & Clements, 2008; Seo & Ginsberg, 2004; Hunting 2010). However, the presence alone of mathematics in play is insufficient for rich learning to occur. Intentional, purposeful teacher interactions are necessary to ensure that mathematical learning is maximized during play (Baroody, Lai, & Mix, 2006; deVries, Thomas, & Warren, 2007; Balfanz, 1999; Ginsburg, Lee, & Boyd, 2008).

Research indicates that supporting the development of young children's mathematical knowledge plays a crucial role in their long-term success in school. In 2007, it was found that mathematics skills among children in Kindergarten were the best predictor of later school achievement, regardless of gender or socio-economic status (Duncan et al., 2007). Further studies confirm this finding (Claessens, Duncan, & Engel, 2009; Claessens & Engel, 2011), and additional work regarding the specific skills needed to be successful indicates that spatial thinking skills and geometric reasoning play a critical role in the development of problem-solving skills, mathematical learning, and reading comprehension (Clements & Sarama, 2011; Wheatley, Brown, & Solano, 1994; Casey et al., 2008).

Numeracy and mathematics share an inherent relationship. On the one hand, such skills as critical thinking and problem solving, applying technology, and understanding the use of data require a solid grounding in mathematical concepts and procedures. On the other hand, knowledge of mathematical concepts and procedures alone is not enough to guarantee numeracy (State of Victoria Department of Education and Early Childhood Development, 2009, p. 6). Some researchers suggest that it is helpful to think of mathematics as a well-established discipline and numeracy as necessarily inter-disciplinary. Steen suggests that numeracy, like writing, must permeate the curriculum. When it does, it will enhance [children's] understanding of all subjects and their capacity to lead informed lives (Steen, 2001, p. 115).

It has also been found that intentionally introducing ideas and materials connected with mathematics in the classroom – or mathematizing the learning environment – can create a wide variety of opportunities for children to learn about mathematics (Clements & Sarama, 2013, p. 136).

Children's prior engagement with mathematics outside the school

Before the onset of formal schooling, young children do not only memorize … and they do not only employ mechanical skills. They deal spontaneously and sometimes joyfully with mathematical ideas. This is what real mathematicians do. (Ginsburg & Ertle, 2008, p. 55)

Young children come to school already knowing a great deal about mathematics. Children bring with them an intuitive knowledge of mathematics that they have developed through curiosity about their physical world and through real-life experiences. It could also be said that, upon entering school, most children are interested in learning to persist, to try something new, and generally to engage in problem solving. Educators play a critical role in fostering a positive attitude towards mathematics by valuing a child's early attempts at problem solving, by sharing and celebrating the child's learning, and by encouraging in each child a love of mathematics.

The following are some examples of ways in which children bring to school their conceptual understanding of mathematics from their daily experiences:

• manipulating objects (e.g., fitting different sizes and shapes of a construction toy together, fitting toys onto a shelf, sorting household items and clothing)
• making comparisons (e.g., “I'm taller than you”; “I'm older than my baby brother”; “I live on the fourth floor and he lives on number ten”)
• making observations (e.g., “This bag is really heavy”; “There are so many rocks”; “I have ten fingers”; “There is a square on my building”)
• asking questions (e.g., “Who is taller?”; “Who has more cookies?”; “How big is it?”)
• solving problems (e.g., “We can make all of these toys fit in this basket”; “Let's see how many steps we have”)

It is essential for Kindergarten programs to build on the prior knowledge and experiences of children. Also, to help maintain the continuity between home and school, educators can encourage parents and other family members to continue to engage their children in similar experiences connected with mathematical thinking at home. Educators can:

• consider including the mathematics involved in the daily school lunch or milk programs in children's exploration of numbers;
• invite children's families to discuss the kinds of mathematics that their children engage in both at school and outside school;
• set up “family math” nights for which children across the school not only plan the mathematics content but also design the program schedule as a mathematics problem-solving task;
• post children's inquiries on the class website or blog, or on a classroom wall or window, and invite parents to respond;
• invite parents to attend a “math circle” at the school;
• create a blog asking a mathematics “question of the week”, such as, “Did you know your children are learning math when they ...?”

Supporting the development of mathematics behaviours

Mathematics in the Kindergarten program builds on children's desire to make sense of their world, and helps them develop and demonstrate their mathematical understanding. Young children use mathematics intuitively and develop their understanding of mathematics through their individual approaches to learning, as well as through their prior experience of their linguistic, family, cultural, and community backgrounds. It is therefore important for children's existing conceptual understanding of mathematics to be valued and for children to be introduced to mathematical concepts in an appropriate manner and at an appropriate time in their development. Children also need to be given learning experiences that are within the range of things they can do with and without guidance, that is, in their zone of proximal development.

When designing learning experiences, educators should consider the children's cognitive, communication/language, physical, social, and emotional development. The most successful learning takes place when the educators provide mathematical experiences that are based on an understanding of the child's total development. The child needs to:

• have the cognitive ability to do the mathematics;
• be able to understand the language of instruction, including the mathematical vocabulary;
• have sufficient fine-motor control to manipulate the materials;
• be emotionally mature enough to deal with the demands of the learning experience so that frustration does not set in.^{footnote 1[1]}

Since all children will demonstrate a developmental progression in the understanding of foundational mathematical concepts, the educators need to ask themselves, “Why have we chosen this learning for this child at this time in this context?”, observe each child, and use their observations to gain insights to negotiate and plan the learning.

The following chart contains some examples of developmental progression in children's understanding of mathematics. Under “Initially”, the examples indicate what a child might say or do at the beginning of the learning process. Under “Eventually”, the examples indicate a more complex understanding of the concept or skill that develops with time and experience.

When children demonstrate knowledge or skills related to such developmental aspects of learning mathematics, they are demonstrating understanding of the seven fundamental mathematical processes. The mathematical processes are described in the following chart (and illustrated with examples in the chart for overall expectation 20 in Chapter 4.5). They can be seen as the processes through which children acquire and apply mathematical knowledge and skills. The processes are interconnected and are integrated with both the overall and specific expectations in the Kindergarten program. The need to highlight these processes arose from the recognition that children should be actively engaged in applying them throughout the program rather than only in connection with particular groups of expectations (e.g., expectations for number sense and numeration, measurement, or geometry and spatial sense). The mathematical processes provide the foundation for mathematical thinking in the Kindergarten program and beyond.

It is important to understand that children are highly capable of complex thinking. In order to avoid limiting the children's thinking, and to help them extend their learning, educators provide challenges that are at the “edge” of the children's learning. An example follows.

This thinking process is not described as an actual expectation for Kindergarten children, but it is a complex example involving understanding of concepts of whole-part relationships, of composing and decomposing quantities, and of interpreting and drawing conclusions from data. It is therefore important to be attuned to children's mathematical thinking – that is, to regard young children as being capable of potentially complex thinking. Keeping in mind the question “Why have we chosen this learning for this child at this time in this context?” will help educators provide differentiated ways of supporting children's learning.

Understanding the importance of connecting mathematics to relevant contexts

Problem solving and reasoning that involve conceptual understandings of mathematics are the foundations of mathematics in Kindergarten classrooms. Rich and relevant mathematical problems involve important mathematical ideas and arise out of real-life situations, and can be approached in a variety of ways so that all children can be involved in exploring solutions. Solving such mathematical problems requires persistence, flexibility in thinking, and multiple perspectives, since there may not be a single, easy-to-find, correct answer. Through mathematics investigations in a wide variety of contexts, children develop their ability to use mathematics as a way of making sense out of their daily experiences. Through these investigations, they also develop increasing confidence along with the knowledge, skills, and attitudes needed to be numerate. Some examples of contexts for investigations are the following:

• in the blocks, sand, or water areas
• at a computer or tablet
• in a small or large group
• during transitions or routines
• in the outdoors

Children can bring more of their current thinking and learning to mathematics materials, questions, and problems when the contexts are relevant and meaningful to them. Attempting to solve problems engages children in posing their own questions and finding a variety of solutions. Throughout the day children should have opportunities to explore and engage in mathematical investigations and to communicate in meaningful ways with the educators and with their peers. When the mathematical problems children are exploring are shown to be connected and relevant to their daily life, the problems provide a vehicle for the children both to apply what they know and to develop new strategies.

A learning opportunity such as the one described above can be thought of as an opportunity for children to engage in a mathematical inquiry if:

• a provocation (e.g., a collection of different kinds of seeds) is provided to engage children's thinking;
• children's questions and theories about the provocation are considered in determining the direction in which their learning will go;
• prompts that continually encourage observation and inquiry are used, such as “What do you notice?”; “What do you think will happen?”; “How could we test your theory?”;
• prompts that encourage making connections between the real world and mathematics are used, such as “Where are other places you see plants growing?”.

The learning opportunity can also make children aware of mathematical processes (see overall expectation 20) if they are asked such questions as the following:

• “How will we show what we found out about our plant's growth?” [communicating; representing; problem solving]
• “In what other ways could we show what we found out?” [communicating; representing; selecting tools and strategies]
• “What did you expect would happen? Why do you think it didn't happen that way?” [reflecting; reasoning and proving; problem solving]
• “How did using mathematics help you learn about the plant?” [connecting; reflecting]

Reading books aloud and in shared reading contexts provides real links between literature and mathematical ideas, since some stories use mathematical terminology and/or contain illustrations of mathematical concepts. Reading can also give children a sense of how mathematics is connected with other aspects of life, such as science and the arts. Because mathematics is potentially relevant and connected to so many areas of inquiry, children should be given many opportunities to demonstrate their understanding in a variety of ways – for example, by constructing concrete models, by describing their understanding in their first language, and/or by making drawings to illustrate a mathematical concept.

The educators can connect mathematics to authentic contexts by:

• developing learning experiences that build on children's intuitive knowledge of mathematics and making use of authentic and culturally and linguistically relevant contexts (e.g., thinking about the role that “number” or “quantity” plays in their lives, such as the number of floors indicated on their apartment building elevator, the number of steps at the front door of the school, the numbers on houses);
• thinking aloud with the children about the mathematical ideas involved when they are putting two blocks together to make one block, making two sides of a painting equivalent, or thinking about how many books will fit on a shelf and why;
• connecting mathematical ideas to literacy, such as patterns in songs, stories, and cumulative texts;
• identifying, exploring, and discussing mathematics in books that they read, in situations that occur in the classroom (e.g., finding ways of making sure that all children have a place to put their boots), and in situations outside the classroom (e.g., comparing sizes of insects found during a schoolyard exploration);
• modelling daily the formulation of mathematical problems, posing questions, and providing opportunities for children to pose questions, and then providing time for investigating possible answers and solutions;
• introducing mathematics into the learning environment – for example, by displaying number lines and hundreds charts; including measuring cups, rulers, measuring tapes, scales, play money, and magnetic numbers; and making explicit connections between using them and doing mathematics.

Mathematics learning throughout the day

The development of understanding of mathematical concepts is not limited to a particular time in the day. Mathematics learning is incorporated throughout the day – it can be made visible, or explicit, to the children in any context, and can be observed by the educators at any time.

Young children have the curiosity and the capability to engage in complex mathematical thinking and learning. Children need to experience mathematics concepts in depth through revisiting and repeating investigations over a long period of time (e.g., the idea of “five” can be represented by the numeral “5” [numerality] to indicate the number of items [quantity] or the fifth person in a line [ordinality]). Enabling children to revisit and think about mathematics in multiple contexts allows their current thinking to be demonstrated and new thinking and learning to be revealed and made visible.

To become proficient, [children] need to spend sustained periods of time doing mathematics – solving problems, reasoning, developing understanding, practicing skills – and building connections between their previous knowledge and new knowledge. (Kilpatrick, Swafford, & Findell, 2001, p. 135)

Educator team members play a critical role in engaging children in mathematics behaviours throughout the day, revealing their thinking and consolidating their learning. The following are some examples of ways in which the educators can engage the children:

• noticing and naming specific mathematics behaviours as they occur (e.g., “Zain, I notice that you used a lot of different shapes to build your structure. How did you decide what shapes to use?”)
• asking the children questions that elicit explanations of their mathematical thinking processes in various contexts
• inviting the children to suggest and negotiate approaches to solving a problem
• encouraging the children to rethink a structure (e.g., offering suggestions for removing or adding materials to make a block tower more secure)
• asking the children to demonstrate, describe, and/or explain their solution to a problem
• helping the children to make connections between what they already know about mathematics and what they are learning at the moment
• identifying mathematical relationships with the children (e.g., two of their small blocks make one large one; different shapes can be combined to make a more complex pattern)
• reviewing children's learning using pedagogical documentation (e.g., watching videos to observe progress), discussing it with the children, and negotiating with the children further opportunities that interest them

Since mathematical concepts are interconnected, learning about relationships between concepts and applications of concepts can take place in multiple contexts. For example, in several expectations related to number sense and numeration (see OE15, and SEs 15.1 to 15.10), key concepts of counting are introduced either as the focus for the expectation (“a number's position in the counting sequence determines its magnitude”, “one-to-one correspondence”, “stable order”, and “order irrelevance”) or in examples (“conservation”, “cardinality”, and “abstraction”). The key concepts of counting are interrelated, and are not necessarily developed in a linear fashion – for example, a child might learn some aspects of one concept, move on to another concept, and then return to work on other aspects of the first concept. Children demonstrate their understanding of these counting concepts in all five areas of mathematics – for example, a child might demonstrate his or her understanding of one-to-one correspondence while analysing data on a graph made by the class.

Throughout the day, the educators can create an effective environment to support young children's learning of mathematics by providing mathematics experiences that focus on particular mathematical concepts and by identifying and embedding significant mathematics learning experiences in play, daily routines, and classroom experiences.

It is important for the educators to use the following approaches when providing learning experiences in mathematics – whether for a large or small group of children or in a classroom learning area or the outdoors. These approaches are interconnected and related to each other, and they are an ongoing part of the educators' interaction with the children. They are not “stand alone” processes nor should they be used in a timed way (i.e., 30 minutes = 10 minutes for each element) or in a linear fashion.

• Observing how children apply their prior knowledge

Educators observe and listen for ways in which children use their prior knowledge to solve a problem, use and manipulate materials, and communicate both verbally and non-verbally. By observing how the children proceed, the educators gain insight into what the children already know, so that they can provide learning opportunities that are challenging but not too challenging (to assess for learning) to ensure that the children will have the necessary tools to develop an understanding of the concept being investigated. Young children reveal their prior knowledge in multiple ways and at different times – not only during an initial encounter with a material or concept. Often children's prior knowledge is revealed in their play and through their actions and conversations, and this unfolding takes place over time and is connected to the learning experience at hand. Educators need to be flexible and to observe continually in order to detect children's prior knowledge and understand how they are applying it in a new context.

• Engaging the children in mathematics

Learning experiences should reflect the children's questions and interests and be embedded in a developmentally appropriate context. For example, with regard to the counting concept that a number's position in a counting sequence determines its magnitude, when a child says “I am four now and I am going to be five”, educators might ask the child, “Are you going to be older or younger on your next birthday?” The learning experiences should also support children in making connections between their ideas/questions/wonderings and the conceptual understandings in the overall expectations. Children need to be able to explore and investigate materials and concepts in concrete ways. By providing opportunities that are interesting to the children, the educators can invite them to engage in such complex explorations as reasoning, investigating mathematical ideas, extending their understanding, reflecting, and making generalizations. Individual learning is supported and extended by both the educators and the children's peers.^{footnote 2[2]}

• Reflecting on the process of children's learning

Reflecting on the process of children's learning creates a mathematics “community” for building understanding of mathematical concepts and thinking about how and where mathematics occurs in our world. The community provides an environment in which individual mathematical ideas can be expressed and tested against others' ideas. ... This enables learners to become clearer and more confident about what they know and understand (Fosnot, 2005a, p. 10). In such a context, educators think flexibly about when, where, with whom, and why to reflect on mathematics. For example, they are attuned to the children's development and think about whether a particular time or place is appropriate for intervention in a child's learning.

Mathematics and the learning environment

Mathematically literate students demonstrate the capacity to “formulate, employ and interpret mathematics” (Brochu et al., 2013, p. 15). They view themselves as mathematicians, knowing that mathematics can be used to understand important issues and to solve meaningful problems, not just in school but also in life. By extension, the physical environment for mathematics learning should include spaces and places for mathematics learning throughout the classroom.