Intentional Teaching Practices in Early Mathematics

An online course from The Education Hub

This is the personal workbook of

Below are your notes and reflections from Part 1 of the course:

Your reflections about maths, prior to the course:

How do you feel about engaging in mathematical activity? This mathematical task made me feel... Would you describe yourself as a mathematician in any way (and if so, which ways?) Why / why not? How confident do you feel about doing mathematics? What experiences do you think have influenced these responses to mathematics?

Notes and reflections on What is mathematics?

According to Melissa Tacy, what is maths? Why do you think children usually start off with great interest in the mathematical patterns and structures that they notice in the world, but are then easily put off mathematics as a school subject? How does this reflection help explain your attitude to learning and using mathematics? Or your experience of doing the quiz at the beginning of this part? What implications do your answers the previous questions have for your philosophy for teaching mathematics?

Notes on culturally responsive mathematics

How does Jodie describe culture and how does this impact culturally responsive teaching?

Note down your response and then read below to confirm.

Final reflection from Part 1

What is your cultural background, and what mathematical practices and activities were you introduced to through your family cultural practices? How well did these align with the way you were taught maths at school? Did this have an impact on your identity as a mathematical learner? Have you ever felt disadvantaged by an incongruence between your mathematical understandings and those promoted by schools and society more broadly? Can you think of examples in which people might be disadvantaged by the privileging of Western mathematics (an example here is the different counting systems of indigenous and non-indigenous Australians)?

Below are your notes and reflections from Part 2 of the course:

Notes and reflections on intentional teaching strategies for supporting children’s mathematical learning

What are the main points that Jodie makes about intentional teaching strategies in mathematics?

Put your learning into practice

In this activity we’d like you to evaluate your current resource provision for supporting mathematical exploration in free play, so that you are making the best opportunities for children’s independent learning. You might like to revisit the sections of the reading above on encouraging number, spatial skills, and pattern exploration in unguided play, and then work through these questions.

What resources do you have in plentiful quantities that children can use to sort, count, and compare magnitudes? Are there large sets of loose parts for this purpose in all learning areas of the early childhood setting? Can some of these sets be sorted according to specific variation in perceptual features such as size, shape, or colour? Are there balance scales or other ways that children can compare the magnitudes of sets of objects? Where are numerals displayed or used in the setting? Can you spot further opportunities to display and use numerals in meaningful ways, such as in dramatic play, or for organisation of resources? Are there number books, charts, or puzzles with written numerals on them? Are board and card games that use numerals available, and how often are these played? What equipment and resources does your setting provide for building and construction play? How do these sets vary? Does the equipment connect, or does it require skills in balance and order? What opportunities are there for children to play with shapes, and are these opportunities both for unguided, creative play with shapes, and for making shapes fit together in a certain way (to copy a picture or complete a puzzle, for example)? What patterns can children observe in the early childhood setting? What resources do you have for pattern-making? Can children work from and copy sample patterns? In what ways are the resources you’ve identified culturally responsive?

Notes and reflections on the intentional maths teaching at Pakuranga Baptist Kindergarten

Notes and reflections on mathematical play

What mathematical skills and concepts are the children exploring in their play? What can you do as a teacher to realise all this powerful mathematical content? Use specific examples from the video and think about some intentional teaching practices that would support children to deepen their mathematical learning, drawing on ideas from Jodie Hunter about drawing attention to mathematical content in children’s play, and Jacqui and her team at Pakuranga Baptist Kindergarten about articulating possibilities for mathematical learning. How aware were you of the different areas that make up mathematics? Which do you think you and your team emphasise most in your practice? Are there any areas that you are less likely to draw attention to?

Below are your notes and reflections from Part 3 of the course:

Notes from Non-symbolic understanding of quantity and magnitude

What does Erica say about children’s intuitive understandings about quantities and magnitude?

Reflections on Non-symbolic quantities in play

What do you think the relationship between non-symbolic understandings of an activity (getting higher on a ladder by taking repeated steps) and symbolic representations of that activity (counting the steps with number names) should be? In other words, when do you think is a good time to start adding symbolic representations to the non-symbolic activity?

Reflections on Building Towers

While this activity is also about learning about shape, size, and the engineering feat of balancing different shapes, in what ways do you see building towers with blocks as building on children’s understanding of quantities? What non-symbolic terms and concepts do you see being used? Now watch the teacher’s reflection. Does she offer any new insights for you?

Reflections on Comparing with glass stones

What would you say is the specific interest of these children in relation to non-symbolic quantities? Now watch the teacher’s reflection. Does she offer any new insights for you?

Put your learning into practice

Let’s now put all this learning about how to support children’s non-symbolic mathematical understandings into practice. There are three parts to this activity: an initial assessment of children’s existing understandings, an intentional and guided activity to promote non-symbolic understandings, and an evaluation or reflection about what you have learned.

Spend part of the day in your early childhood setting paying close attention to the ways in which children are exploring non-symbolic understandings of mathematics in their play. You can do this on a more general level (identifying opportunities at snack-time, or in different areas of play), or you can observe a group of children and the non-symbolic mathematics that is apparent in their play interests.

Try to come up with a list.

Depending on your confidence with this area of mathematics, either choose one of the following activities to carry out with a child or group in your setting (detailed instructions are available for each by clicking on the link), or build your own intentional activity with a focus on non-symbolic understandings by downloading and then filling out this template.

a. Sorting and building. Use sorting, grouping, building, and comparing to explore non-symbolic number understandings (aimed at infants and toddlers).

b. Comparing with loose parts. Play with loose parts to practise sorting and comparing magnitudes before doing something creative with them (aimed at younger and older children).

c. Reading The Hungry Caterpillar before snack time then compare quantities of food at snack-time (aimed at toddlers and younger children)

How did the children respond? What do you think the children were learning? Did this activity build on their existing mathematical skills? In what ways? What did you learn about the children’s mathematical knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 4 of the course:

Notes on Developing symbolic number knowledge and skills

Why does Erica believe number symbols and number skills are so important?

Reflections on Number songs

What kind of concepts are the toddlers being exposed to? How might sharing number songs support their learning about number? What other mathematical learning do you see here? Would you have done this activity any differently? Now watch the teacher’s reflection. Does she offer any new insights for you?

Reflections on Making playdough from a recipe

How does having the recipe for playdough written out support children’s mathematical understandings? What different kinds of number are put to use here? What other mathematical skills come up as Nicola and the children follow the recipe? Now watch the teacher’s reflection. Does she offer any new insights for you?

Put your learning into practice

Time to explore what symbolic number skills children in your setting have, and try out some of our activities. There are three parts to this activity: assessing children’s existing understandings, an intentional and guided activity, and an evaluation or reflection about what you have learned.

To find out what children know about symbolic number, spend time observing the ways in which children use number symbols in play. You might like to set up a ‘checking for understanding’ activity in which children use number symbols to direct their activity – for example, picking a numeral card and reading it out in the context of a game, or counting by rote while skipping, or playing hide and seek. How many number names do children know, and can they get the order right? Do they know any numerals? Note that we are not concerned with accurate counting here, which is covered in Part 5, but just knowing the number names and their order.

Choose one of the following activities to carry out with a child or group in your setting (instructions via the link), or, if you are feeling more confident, build your own intentional activity with a focus on symbolic number skills by filling in this template.

a) Number songs. Sing songs to provide children with opportunities to hear number names used in meaningful contexts (aimed at all ages).

b. Following a recipe. Use your favourite recipe, or the one provided for making playdough, to practise counting, comparing, and ordering when cooking (aimed at younger and older children).

c. Maths obstacle course. Set up a numbered obstacle course, and practise talking about numerals and the way in which numbers help us put things in order (aimed at younger and older children).

Reflect on how this activity went:

How did the children respond? What do you think the children were learning? Did this activity build on their existing mathematical skills? In what ways? What did you learn about the children’s mathematical knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 5 of the course:

Notes and reflections on Connecting non symbolic and symbolic number knowledge

When it comes to learning colours and numbers, most adults seem quite comfortable with quizzing children: ‘What colour is this?’ or ‘How many are there?’ These questions are clearly designed (and likely interpreted by children) as questions to test children’s knowledge and understanding.

Why else would an adult ask a question they already know the answer to? What do you think about this? What image of the child (and adult), and relations between children and adults, do these questions create? What are some of the strategies we can use to ensure counting is meaningful and authentic for children? If we do want to know how well children can count, or if they know something (because this is important information for teachers that helps us know how to support children better), how might we frame it in ways that preserve respectful and empowering relationships between children and teachers?

Reflections on Counting difficulties

What do the children know about counting? Try to be specific about this, as well as the particular difficulty they are having, relating these to the skills of knowing the count string, one-to-one correspondence, and cardinality.

Reflections on Creative number tiles

What opportunities does an exploration of quantities present for also exploring space and shape? Now watch the teacher’s reflection. Does she offer any new insights for you?

Reflections on Beanbags in boxes

What skill do you think the teachers were trying to intentionally promote by the use of the patterned dots on the box lids? Now watch the teacher’s reflection. Does she offer any new insights for you?

Put your learning into practice

Here is your opportunity to bring together all the information provided about how children learn to connect symbolic number with non-symbolic quantities, in order to support children’s mathematical learning in your setting. This involves three activities: an initial assessment of children’s existing understandings, an intentional and guided activity to promote children’s understanding of the connection between symbolic number and non-symbolic quantities, and an evaluation or reflection about what you have learned.

Spend part of the day in your early childhood setting paying close attention to the ways in which children are making connections between symbolic number and non-symbolic quantities in their play, through counting, subitising, and numeral knowledge. You can do this on a more general level (identifying opportunities during different daily routines), or you can observe a group of children and see the ways in which they use counting and numerals in their play. Try to work out which of the skills for counting and connecting numbers with sets of objects children have mastered, and which are not apparent yet, or less secure.

Depending on your confidence with this area of mathematics, either choose one of the following activities to carry out with a child or group in your setting (detailed instructions are available for each by clicking on the link), or build your own intentional activity with a focus on connecting symbolic number names or numerals with non-symbolic quantities by filling in this template.

a) Help a puppet learn to count objects, with the children identifying and helping to correct their mistakes (aimed at toddlers and younger and older children)

b. Outdoor number line for children to travel up and down according to the roll of the die (aimed at younger and older children)

c. Beanbags in boxes game, as in the video (for younger and older children)

d. Memory game matching dot patterns to numerals, with possibilities to extend into making own memory cards (aimed at younger and older children)

Reflect on how this activity went:

How did the children respond? What do you think the children were learning? Did this activity build on their existing mathematical skills? In what ways? What did you learn about the children’s mathematical knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 6 of the course:

Notes on Comparing and combining with symbolic numbers

What ideas does Erica have for teachers to support children’s understanding of symbolic relations and combinations

Reflections on the card game War

What other mathematical skills, beside symbolic magnitude comparison, do you see Beth using as she plays this game with Marilyn? Now watch the teacher’s reflection. Does she offer any new insights for you?

Reflections on Café roll play

In particular, how did she connect number symbols to non-symbolic quantities to help the children? What other ways (use of language, use of props) can you think of that would support children to translate symbolic operations into non-symbolic quantities, and back again? Now watch the teacher’s reflection. Does she offer any new insights for you?

Put your learning into practice

Now it is your turn to explore symbolic magnitude comparison and symbolic arithmetic in your setting. As before, there are three parts to this activity: assessing children’s existing understandings, an intentional and guided activity, and an evaluation or reflection about what you have learned.

To learn about the children’s understanding of symbolic magnitude comparison and/or symbolic arithmetic, set up a couple of scenarios. You might use a game or a storybook. For symbolic magnitude comparison, you can hand out numeral cards and ask who has the largest (put this in a fun context, such as determining who is going to sit at the table for morning tea first, or challenge the children to do something physical like jump the number on their card). For symbolic arithmetic, you might use some of the food available at morning tea, and ask, for example, what three slices of apple and two raisins makes (children can physically count out apples and raisins to find the answer). You might introduce an arithmetic problem using puppets or toys that you specially introduce (‘if monkey wants three bananas and mouse wants one…’), or use some of the materials children are playing with to create arithmetic problems. However, be really careful not to interrupt meaningful play to interject with a maths question, as this won’t build positive attitudes to maths!

How did you get on?

Choose one of the following activities to carry out with a child or group in your setting (instructions via the link), or, if you are feeling more confident, build your own intentional activity with a focus on either symbolic magnitude comparison or symbolic arithmetic by filling in this template.

a. A teddy bear’s picnic to practise comparing symbolic numbers (aimed at younger children)

b. Role play shop or café to practise comparing and combining symbolic numbers_ _(aimed at older children)

c. War card game to practice symbolic magnitude comparison (aimed at older children)

Reflect on how this activity went:

How did the children respond? What do you think the children were learning? Did this activity build on their existing mathematical skills? In what ways? What did you learn about the children’s mathematical knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 7 of the course:

Notes on spatial skills

What are the links that Erica describes between children’s spatial skills and patterning skills and their general mathematical achievement?

Notes and reflections on shape heuristic play

Which objects do you notice that Lottie is most interested in? Do you think she can discriminate between the different shapes? What makes you think so? What mathematical dispositions can you see Lottie developing? Now watch the teacher’s reflection. Does she offer any new insights for you?

Notes and reflections on Shape play

How would you describe the teaching strategies Merryn uses here? What opportunities did she provide for children to develop the different kinds of spatial skills: form perception or shape knowledge, visual-spatial working memory, and spatial visualisation? Now watch the teacher’s reflection. Does she offer any new insights for you?

Notes and reflections on Shape corner game

In what ways did Jess use the game to teach children different shape vocabulary and concepts? How might this game be used as an assessment tool? How might you adapt this game to teach different content? Now watch the teacher’s reflection. Does she offer any new insights for you?

Put your learning into practice

Next, we want you to explore one aspect of children’s spatial skill development in your own setting, and plan how to support it further. There are three parts to this activity: an initial assessment of children’s spatial skills, an intentional and guided activity to promote and extend children’s spatial skills, and an evaluation or reflection about what you have learned.

Spend part of the day in your early childhood setting paying close attention to the ways in which children are exploring and making use of spatial skills. You can do this by spending time in one area, such as the block area or sandpit, and observing how children draw on spatial skills there, or you can observe a single child or group of children and see the ways in which they use spatial skills across a range of areas of play within a limited time period.

It might be best to produce a simple bullet-point list.

Depending on your confidence with this area of mathematics, either choose one of the following activities to carry out with a child or group in your setting (detailed instructions available for each by clicking on the link), or build your own intentional activity with a focus on spatial skills by filling in this template.

a. Treasure basket and shape sorter boxes. Explore shapes through heuristic play (aimed at infants and toddlers).

b. Collage with shapes. Play with shapes to create 2D shape pictures (aimed at toddlers, younger and older children).

c. Corners game. Practise shape recognition with a fun and active music game (aimed at toddlers, younger and older children).

d. Signature dishes (restaurant role play or at snack time). Practise visual-spatial working memory by recreating the chef’s signature dish (aimed at younger and older children).

Reflect on how this activity went:

How did the children respond? What do you think the children were learning? Did this activity build on their existing mathematical skills? In what ways? What did you learn about the children’s mathematical knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 8 of the course:

Notes on Patterning skills

In what ways does Erica say that patterning skills support general mathematical skils?

Notes and reflections on patterns

The text describes the different stages of learning about patterns. What are the stages that are mentioned? Think about a particular child in your centre. What stage are they at? How did you come to that conclusion? The text includes ideas about how to teach pattern skills. Thinking about your specific child again and where they currently are with their patterning skills, what activity might you do with them? What challenges might you expect to encouter with this specific child and the activity you have chosen? How might you mitigate those challenges?

Put your learning into practice

This time our practical activity is focused on finding out what patterning skills children in your setting have, and on trying out some of our activities. As before, there are three parts to this activity: assessing children’s existing pattern understandings, an intentional and guided activity, and an evaluation or reflection about what you have learned.

For this activity, spend time learning about what kind of patterning challenges children can do. Here you will set up a ‘checking for understanding’ activity in which children either copy, fix, extend, or abstract a pattern. If children can complete the first kind of patterning skill, move onto the next.

How did you get on?

Choose one of the following activities to carry out with a child or group in your setting (instructions via the link), or, if you are feeling more confident, build your own intentional activity with a focus on pattern, by filling in this template. All of these activities can be geared to copying, fixing, extending, or abstracting patterns as appropriate for children.

a. Pattern dancing. Watch an example song from GoNoodle on YouTube and have fun (for toddlers, younger and older children).

b. Pattern plates. Have fun creating snacks with repeating patterns. The children’s book Pitter Pattern in our further resources section has a page in which snacks are arranged in a pattern which might be used as a stimulus for this activity (for younger and older children).

c. Pattern creations. Use a variety of materials to create patterns (for younger and older children).

Reflect on how this activity went:

How did the children respond? What do you think the children were learning? Did this activity build on their existing patterning skills? In what ways? What did you learn about the children’s patterning knowledge? How will you build on the learning you noticed? Which children might you choose to follow up with and how?

Below are your notes and reflections from Part 9 of the course:

Notes on mathematical play

Is the mathematical content of children’s play more obvious, or are you noticing different things? Are you looking at learning much more through a mathematical lens? Do you see lots of potential for ‘mathematising’ this play? Does the mathematical content of children’s play excite you?

Watch the video one more time.

Can you imagine what you might say or do to emphasise or reinforce the mathematical concepts children might explore?

Final thoughts

Has this changed as a result of your learning on this course? How do you feel about doing mathematics now, and particularly doing mathematics with children? What kinds of attitudes and feelings come up when you think about doing mathematics? Do you feel more of a sense of competence? How would you describe yourself and your ability in regard to mathematics now? What impact do you think this might have on the children you teach? Finally, what can you do personally to continue your professional growth in mathematics?

Mathematical project

What opportunities might there be for you to build or introduce a mathematical focus into a new or existing project or interest of the children in your centre? Brainstorm some ideas to take to your teaching team, and to discuss with children and families.

If you are keen to get started, you might find this planning proforma useful.