Developing the ability to play and experiment with numbers flexibly

Developing a deep understanding of number and being able to use numbers flexibly in any given problem or situation is referred to by Boaler as ‘number sense’. Deep number sense is essential for creatively manipulating numbers in order to solve maths problems in and out of the classroom. To read more about this and to watch videos of this in action visit the YouCubed website: https://www.youcubed.org/resource/number-sense/

As teachers we need to balance the need for developing fluency, conceptual understanding and problem solving skills during our lessons. Boaler discussed that without this balance in Mathematic lessons, students become rote learners who memorise rules, have low number sense and an inability to flexibly use numbers. Below is an example of a balanced and unbalanced learning triangle that was presented at the Mathematical Mindset workshop (May 2019, Jo Boaler, Stanford University)

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Within a single Year 5 lesson at The Australian International School, Singapore, the teacher addressed all three aspects of Mathematics that Boaler ascertains as vital. The teacher commenced the lesson with a fractions ‘Number talk’, followed by a multiplication game to practise fluency, which lead into a problem solving task that incorporated both concepts. The Numeracy leader at this school discussed the shared belief that even though fluency of Maths was important, it did not equate to speed, 100% accurately or solving problems in only one way. Boaler discussed her own inability to learn the times tables in primary school, which many people believe is vital for success in Mathematics, however, she could solve multiplication sums by flexibly using known facts and her understanding of numbers.

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  Number talk task in Year 5.

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  During the number talk students explained their thinking verbally, through drawings and with equipment.

When students are able to use numbers flexibly it demonstrates a deep conceptual understanding rather than a procedural understanding of numbers that only requires recall of facts and rules. During a classroom visit I observed Year 1 students with low number sense and an inability to flexibly use numbers when they were completing addition worksheets. As part of the school routine, students completed worksheets daily to increase fluency. Students were solving every single addition question the same way; counting all, using their fingers. The students did not have a conceptual understanding of numbers or addition that could help them solve the problems efficiently using a variety of strategies.

This observation was juxtaposed to another Year 1 class at The Australian International School, Singapore, where Year 1 students were completing subtraction problems by flexibly experimenting with numbers and various effective strategies (such as counting on, ‘friends of 10’, doubles, the commutative property or building to the next 10). The students were also experimenting with how to represent their thinking; by using different equipment, pictures and numerical form.

While you are planning your Maths lessons for next week see if you can do these things each lesson and reflect on how your students work, how they think and how they develop Maths content knowledge:

• Can you include a fun way to increase fact fluency (think fun games rather than timed worksheets…don’t forget teacher questioing can transform fun games to deep thinking learning experiences)?

These students are increasing fluency of addition strategies by playing ‘Friends of 10’ memory / concentration.

• Can you include some number talks to increase conceptual understanding of a concept (put up a sum that students may not be able to automatically solve…18×5 (Stage 2 or 3 students ) or 6 + 4 + 6 (Kinder or Year 1 students)…ask students to discuss all the different strategies they can use to solve it).

This number talk was conducted in a Year 5 class to provoke discussions about fractions, equivalence and fraction relationships.

• Can you include a rich problem solving task that is ‘low floor-high ceiling’ that also integrates more than one number strand…this will allow all students to access the task but be open enough for students to extend and challenge themselves to think creatively and critically about all the interconnected concepts involved.

‘Imagine you have 16 children sitting in rows at the movies. What equal rows could they sit in?’ Challenge: What about 32 children? (NB: the Year 2 child discovered the connection between 16 and 32 after their first example and then extended their thinking to more related numbers…64, 128 etc)