Children can spend a lot of time in mathematics lessons learning about and practising procedures. This is understandable as procedures are very powerful and can provide a quick route to finding answers. However, this can give the impression that performing procedures is what mathematics is all about, to the point where children stop making sense of the mathematical elements (eg numbers) that they are working with. As an extreme example, I have in mind a Year 7 child who, when asked to find 20+20, instead of coming up with 40 in their head, felt compelled to write the numbers in a column: "0 plus 0 is 0, 2 plus 2 is 4, the answer is 40".
This is not an argument against procedures, but about the dangers of giving them too much emphasis, and at too early a stage. Schools, teachers and pupils are put under considerable pressure these days by the high-stakes tests that pupils are required to take. In the UK, this applies especially to the Key Stage 2 national curriculum tests (SATs or Standardised Assessment Tests) that pupils take towards the end of Year 6 (age 10-11 years). For mathematics, pupils take three such tests, one on 'arithmetic', two on 'reasoning', and there is a strong focus on procedures in all of them.
The aim of the tasks in this blog is to empower children. It might be argued that teaching powerful procedures is precisely the way to do that, but that only works if children understand (or are close to understanding) them. Without that, the sense of helplessness that the rote use of procedures can engender is likely to be disempowering.
The tasks are designed to help fill the gap that a focus on procedures might have produced. It is hoped that the tasks will tap into children's ‘natural’ intelligence. Think of all the things that a 10 year old, say, has achieved and can do - speak a language, run across a field, walk without bumping into things, catch a ball, climb stairs, read and write, interact with people, play in a team, ride a bike, swim, watch television, do productive things on a mobile phone... I want children to realise that they have the capacity to make sense of maths, to engage in mathematical thinking and to grow in their understanding, competence and enjoyment, whether or not they can cope with formal procedures.
Some of the tasks in this blog will use items from the 2025 Key Stage 2 SAT arithmetic test to explore other, mathematically interesting, ways in which the items can be solved. But the blog will mostly contain non-standard tasks so that children who would normally reach for a procedure are given the opportunity to use some of the many skills that they have developed as they grow up.
For
convenience, I have arranged the blog into sets of 'weekly' tasks,
usually with 5 tasks per week. However, this format should not be taken
too literally. For example, it doesn't mean that you should use one task
every day, or only one task, and I would urge that when you use a
task, allow plenty of time; don't rush through it and on to the next. It
also doesn't mean that the weekly sets, or the tasks within them, have
to be used in the order they are presented here. The order is up to you.
In sum, my hope is that the tasks in this blog will help children get a better feel for what mathematics and mathematical thinking is about and that this will help them develop or maintain the habit of sense-making so that they are confident about maths and enjoy it.
If you have any comments about the blog, or would like to share experiences of using the tasks, please email me here: mathsuntangle25@gmail.com
Dietmar Küchemann