The post yesterday was related to making sense of part-whole addition and subtraction strategies that pupils use and showed some examples of these with concrete materials and visual representations as well as exploring the mathematical properties that underpin them.
The guidance document I’m going to link to today has a progression of number ranges at First** and First***. The two documents could be used alongside one another to support your planning. (We will hopefully create one of these for Second Level too, in due course.)
An overview of the document is provided below including some important points to keep in mind.
- The numbers are all presented as bare numbers e.g. 37 + 8 (rather than in word problems or other contexts) this is purely to illustrate the numbers involved and not a suggestion that you should only provide examples in this way.
- These numbers should be used as part of a variety of problem types as has been exemplified elsewhere, these include problems that involve comparison (and looking at the difference), situations where you are breaking a whole into parts (part-part-whole) and problems that involve the more common problem types of increasing or decreasing a quantity including varying which part of the problem is unknown.
This also includes varying the type of practice task you give pupils. Presenting tasks in different ways will help draw out different relationships or underlying properties.
- When you use the number ranges, they should include a variety of equation representations e.g. 15 = 5 + ? or 6 + ? = 5 + 10 as well as the standard representation 5 + ? = 15. This helps pupils make better sense of the actual meaning of the equal sign. This should also extend to the use of inequality symbols e.g 5 + 10 < 6 + ?
- At each level there are several main milestones, for want of a better word, in terms of solving a particular type of problem. These have been broken down into smaller steps but this is not necessarily so that teachers teach in these micro-steps one step at a time. It’s more to help teachers understand the smaller skills involved in achieving the end goal. They can then use this to help them make sense of how they are going to structure their teaching, what type of practice type might be useful, what connections they will make and how they will support pupils to understand the mathematical properties.
- And last but not least, there is a summary of the types of numbers that lend themselves to the different strategies that were explored in the guidance document that was explored yesterday.
From my experience, a lot of textbooks don’t promote this level of thinking and the way the questions are organised and the types of representations they offer often limit the opportunities for really understanding and exploring the mathematics in an enjoyable way that promotes pupil questioning and curiosity.
Hopefully this document alongside the other ones we are sharing will mean people feel better equipped to understand the progression of learning and also plan meaningful, engaging experiences for their pupils.
As always, if anyone spots any mistakes or disagrees with anything then please do let us know!