Mathematical Thinking in Kindergarten

 Mathematical Thinking

Let me start with a series of engagements from my Kindergarten class in the past week as we explored the process of representation using the context of number. 

We started with a read aloud of One is a Snail, Ten is a Crab by April Pulley Sayre and Jeff Sayre. The book uses animals' feet (and their combinations) to represent different numbers. The children were then invited to choose their own representations of 1 and 10 initially, and then other numbers. Although all children used a range of representations, they all showed 10 as a collection of ten of whatever they had chosen to represent 1. 

The next day, to help the children think mathematically about efficiency in representations, we used the Smartboard to represent numbers in two teams - the "individual squares" team and the "crabs" team. On rotation, one member of each team raced each another to represent numbers such as 60. The children had opportunities to communicate what they noticed, make connections and explain their ideas. Imagine the thinking when the number to represent was 3! I helped the children to construct their own meaning and think for themselves through careful questioning. There were many working theories we tested out and explored together, including the idea that the representation for 10 had to bigger in size than the representation for 1. At this point, I asked the children to imagine the letter C to represent 100. The children laughed! I continued, playfully, and said X is going to represent 10, V will represent 5 and I will represent 1. They told me it wasn't possible, and I said in this system it was! Like the book that combined animals, I invited the children to think about the value of representations such as XXI. This exploration of Roman numerals was spontaneous and in response to the thinking of this particular group of 5 and 6-year olds. 

In the next maths lesson, we reviewed representations with animals, shapes and Roman numerals. Using ideas from Jo Boaler's Mindset Mathematics Grade K, the children had opportunities to count large amounts of loose parts in the classroom (mosaic tiles, bottle tops, dominoes) and think about ways to organise the materials to count them efficiently. Following this, the children were asked to represent different 2-digit numbers (eg. 72) using loose parts, thinking about the organisation of the materials to ensure the representation was accurate and efficient. Each number was written on the board with digits, noticing that this was also a representation of number. The connections children were making between these different experiences were so rich. 

Investigations for the coming weeks include representations such as Numicon, 10 Frames, number beads, Rekenreks, number lines, 100 squares, place value arrow cards and base 10 Dienes materials. 

This weekend, I saw a post from Dr Helen Williams on social media which said: A worksheet - even on a PPT (PowerPoint) - is not a "maths task". Today's thought. It made me wonder why, when given a choice, worksheets with low-level and repetitive questions are used over rich learning opportunities to engage children in mathematical processes. This post is an effort to share some of my own thinking and a set of resources I have curated to place a spotlight on what mathematical thinking is, the pedagogies it is part of, and some strategies and resources that might inspire teachers with ideas about how one might put thinking at the heart of a maths curriculum. 

Alongside understanding the what of maths (place value, probability, length etc), there should be a spotlight placed on the how of maths which supports mathematical thinking and can be thought of as a set of processes. These transcend the what of maths, as well as transcending age, too. They are as relevant to 4-year olds as they are to 14-year olds. 

I have unpacked the mathematical processes listed from the NCTM. Perhaps you might use this to reflect on your own classroom practice to celebrate strengths and identify some aspects of processes to develop.

These processes are nurtured in pedagogies and a culture of thinking that value:

- play and playful learning where open-ended, mathematical explorations germinate in both child-led and adult initiated learning. Further practical examples of Kindergarten mathematics can be seen in a previous blog here. 

- conceptual learning through which children develop an understanding of big ideas, alongside the development of important skills and knowledge. Mathematics is inherently a conceptual discipline, but it is important that concepts like place value, estimation and symmetry are explored and unpacked over time and in different contexts so that depth is not sacrificed for "quick fix" coverage. 

- a growth mindset where mistakes are valued as opportunities to learn. 

- authentic learning contexts so that children see the relevance of what they are learning in their everyday lives. We often use the outdoors, different parts of the school and Paris as a classroom so that children can apply the real world to mathematics, and mathematics to the real world. An example of contexts that might be used to spark interest in or connections to shape and space can be found here. 

- inquiry through which children are actively involved in the exploration of ideas that they feel strongly connected to, curious about, intrigued by and have a desire to make sense of. This is a graphic that we use in Kindergarten to help children visualise inquiry as a journey, inspired by the work of Kath Murdoch and Jo Boaler. 

- mathematizing! What a great word to capture learners engaged as mathematicians. What connections can you make to the mathematical processes?

What are some practical and concrete examples of ways to engage children in mathematical thinking?

- Loose parts (including, but beyond what might be considered 'manipulatives' for mathematics) so that children can visualise and represent abstract concepts in concrete ways

- Games

- Challenges

- Literature

- How many ways can you...?

- The answer is ___. What might the question be?

- Give children a generalisation. Is it true or false? Show me. 

- Show children several examples of (graphs, shapes, numbers, patterns...) Generalise what they have in common. 

- Take a walk (around school, in a garden, to the park, at home, down the street...) to observe, notice and wonder

- Show children different statements to classify based on if they are ALWAYS, SOMETIMES or NEVER true. 

- Number talks including Number dots

- Which one doesn't belong? 

- Reflecting on thinking as the processes are being used 

The types of questions that we ask reveal so much about what we value in learning. Here are some potential questions to play with that might elicit a higher level of thinking more aligned with the mathematical processes. 

What are some resources to find further ideas and inspiration?

NRICH Examples of conceptually rich opportunities

YouCubed Visual, creative conceptual maths opportunities

Visible Thinking Routines from Project Zero, Harvard

Thinking Moves A to Z A set of thinking moves that can be used with any ideas (including mathematical ideas)

Cultures of Thinking by Ron Ritchart

Blog by Dr Helen J Williams

Ways to encourage deeper thinking inspired by Andrew Jeffrey

Confront the unusual

A Padlet of curated resources

Which mathematical processes do your learners show evidence of? Which ones might you focus on next?

What strategies do you use to engage children in mathematical thinking?

What resources do you find inspiring to develop mathematical thinking? Please share!