This is the second in a three-part series that I have developed when I have been thinking about Mathematical Fluency. Fluency in Maths has been highlighted as an aim in the National Curriculum and it is down to us as educators to ensure children are equipped with the tools needed to access such fluency. Last week I posted about the power in children teaching others to develop their fluency. This week I will focus on building their understanding of concepts and in the final week I will unpick problem solving.
2. Conceptual Understanding
When we teach children methods in Maths, there is a danger that we overlook teaching them why we do certain things. A classic example is teaching a written method for addition. When children are eventually taught the standardised column method (as in the Appendix of the National Curriculum, following on from non-formal methods such as the number line) they are taught to ‘carry over the one’ or some other vague comment meaning we carry a remainder over from the previous place value. Do all children understand that ‘one’ is actually a hundred being carried over from the addition in the ten column? Maybe, maybe not. It is such conceptual understanding that is vital in developing the mathematical fluency in a child’s knowledge of working with number.
Recently, as mentioned last week, our school had an Ofsted inspection. In a discussion with Year 5 pupils, the understanding of this sign was brought up ‘=’. The children were fine with this (x+5=9, what is x?) but there was slight confusion when this problem was shown (x+5=6+y – what is the value of x and y?). These children, according to National Assessments, were competent mathematicians. The problem was not in being able to ‘do’ Maths but in ‘understanding’ – that ‘=’ doesn’t just mean ‘makes’ or ‘comes to’ but literally means ‘is equal to’. Our school has an extremely high proportion of children who speak English as an Additional Language so it may come as no surprise that the most challenging area in Maths might be in language and terminology rather than in ‘doing’ the Maths.
How do we help develop children’s conceptional understanding rather than just training them in the ability to go through the mechanics of methods? There will be a number of ways. Recently, my wife became an Usborne Independent Organiser. Basically she promotes a love of reading through organising parties based around the Usborne Book Publisher and tries to generate interest. In the Beginner Pack she received, there was a ‘First Illustrated Maths Dictionary’. See link below:
This was the first I had heard of a ‘Maths Dictionary’ (and this post is not to sell the book to you, I’m sure many other Maths Dictionaries are available – although if you would like a copy then let me know ;P)
Having had a look through it, I thought it was a brilliant book! Very colourful, engaging and goes through concepts found in the National Curriculum. There is also a 7+ version and 11+ version. These publications go through the language used in Maths (including the ‘=’ sign mentioned before) as well as many other mathematical concepts. I think this is another medium through which we can try to develop children’s mathematical fluency by consolidating their conceptual understanding.
Are there any other publications that you are aware of that could support children’s Maths understanding? It is pretty clear that if we develop children’s conceptual understanding then this will improve their fluency – but do you have any ideas or techniques that have worked in the classroom?