Haylock and Cockburn

My apologies to any readers of this blog for a long period of silence. Anne Cockburn and I have been busy working on a fifth edition of our Book, Understanding Mathematics for Young Children. We finished this last week and have sent off the 'manuscript' to the Sage Publications.

The word 'Manuscript', which is still used in publishing circles, means, of course, 'written by hand', which is strangely archaic, given that no pens and no paper were involved in either the writing process or the submission of the new edition to the publisher.

We hope to see the new edition on the shelves (another anachronism, since most of the sales will be online!) in the first half of next year.

We have had to rework the book to ensure that it is consistent with the language and content of the new primary mathematics curriculum in England. This has introduced some new material for Year 2 children, such as fractions. The new curriculum has meant a general shift of content down from Year 3 to Year 2 (and likewise from Year 4 to Year 3). So, to avoid our book getting even longer, we have had to make a decision to reduce the age range covered. So, it is now described as 'a guide for teachers of children aged 3–7 years' (rather than 3–8). This is a better fit for teacher training courses anyway, since it is now clearly aimed at Early Years Foundation Stage and Key Stage 1.

So, with that done, I hope to get back to writing the occasional blog again! Watch this space.

Mastery and understanding mathematics

In the context of the challenge to raise standards in mathematics in schools in England the word ‘mastery’ has recently become prominent in the vocabulary of the English mathematics curriculum (NCTEM, 2014, www.ncetm.org.uk/public/files/19990433). It is reassuring to note that the way in which the word ‘mastery’ is being used is entirely consistent with the approach to children’s learning of mathematics that I have promoted in my own writing.

Mastery is seen as children developing fluency in mathematics alongside a deep understanding of mathematical ideas and processes. So, for example, teaching approaches for mastery should ‘foster deep conceptual and procedural knowledge’ and ‘exercises are structured with great care to build deep conceptual knowledge alongside developing procedural fluency’ (op.cit.). This is a key principle in teaching mathematics to young children: that mastery of the subject is not achieved simply by repeated drill in various procedures. Instead, the focus is on the development of understanding of mathematical structures and on making connections.

Making connections in mathematics – a recurring theme in all my books – ensures that ‘what is learnt is sustained over time, and cuts down the time required to assimilate and master later concepts and techniques’ (op.cit.). Nearly all mathematical concepts and principles occur and can be applied in a wide range of contexts and situations. Because of this, the deeper understanding central to mastery in mathematics is facilitated by a wide variation in the experiences that embody mathematical ideas.

For example, mastery of the 5-times multiplication table by Year 2 children is not just a matter of memorizing a chant that begins ‘one five is five, two fives are ten …’ – although that is part of it. It would also involve, for example:

·       connecting each result in the table with a collection of 5p coins and the total value;

·       articulating the pattern of 5s and 0s in the units position in the odd and even multiples of 5;

·       explaining how to get from 4 fives to 8 fives by doubling;

·       explaining how to get from 6 fives to 7 fives by adding 5;

·       counting in steps of five along a counting stick;

·       knowing that, say, ‘3 fives are fifteen’ is what you use for the cost of 3 books at £5 each;

·       constructing patterns with linked cubes that show 1 set of five, 2 sets of five, and so on;

·       filling in the missing number in number sentences like ‘6 × □ = 30’.

To teach for this kind of mastery teachers themselves need a deep structural understanding of mathematics, an awareness of the range and variety of situations in which a mathematical concept or principle can be experienced, and confidence in exploring the connections that are always there to be made in understanding mathematics. Any teachers looking for this? I can recommend one or two books.

"By George, she's got it!"

These are the words of Professor Higgins in My Fair Lady when Eliza shows significant progress in her learning! We use a lot of different idioms to indicate that we understand something or do not understand something. For example, for understanding, we might say things like:

• Oh, I see! 
• Now it’s clicked. 
• Everything is falling into place. 
• I think I've cottoned on! 
• I get the picture. 
• My eyes have been opened! 

And for not understanding, we have phrases like these:

• Sorry, I don’t get it. 
• I’m still in the dark. 
• I can’t see the sense in that! 
• That's as clear as mud! 
• You're talking double Dutch! 
• It just went over my head. 

What's your favourite way of saying 'I understand' or 'I do not understand'?

Expressions like those above reveal a number of insights into the nature of understanding. They show, for example, the importance to us of understanding things, of making sense of them, rather than just learning by rote. We can see within the expressions used a clear sense of closure, of things fitting into place, a sense of relief almost, when we 'get it'. And, by contrast, phrases that indicate not understanding reveal a sense of frustration. Understanding or not understanding is as much an emotional experience as a cognitive one!

Then we might notice that understanding brings clarity and light to the learner. Many of the phrases we use for understanding talk about 'seeing' something, having our eyes opened, or, by contrast, being in the dark.

Finally, these expressions for understanding support the idea that learning with understanding is all about making connections: connecting some new experience with other experiences or existing understandings. So, we talk about understanding as though it feels like things 'clicking' (like two Lego pieces fitting together), or 'falling into place' (like a piece in a jigsaw connecting with other pieces and suddenly making sense), or 'getting the picture'. And when we don't understand, the words 'go over our head', so we have a sense of not being able to connect them to anything and embed them in our minds. 

To learn that mathematics can be learnt with understanding in the ways indicated by these idioms is the most important thing for children to learn about this subject by the time they leave primary schools.

New edition of Haylock & Cockburn

There's the usual excitement at the Haylock household in Norwich to greet another publication: a new edition of Understanding Mathematics for Young Children, written by Anne Cockburn and me. Below is the link to the Sage website for more details.

http://www.uk.sagepub.com/books/Book237964?siteId=sage-uk&prodTypes=any&q=haylock&fs=1

We have lots of new material in this new (fourth) edition. We have drawn especially on Anne's extensive involvement in international mathematics education organisations to include a Research Focus at the end of each chapter. For example, in the chapter on Understanding Calculations through Patterns and Pictures the Research Focus describes some of the research into children's use of empty number lines for subtraction calculations. Then we also include here and there a Pause for Thought, which is an opportunity to do just that. For example, in the same chapter after explaining how to use hundred squares, all starting with 1 in the top-left hand corner, in a Pause for Thought we ask why it might be that some teachers argue that hundred squares should start with zero rather than 1, and others that we should start counting on a hundred square from the bottom left-hand corner rather than the top.

What else? Well, the book is in full colour on very nice quality paper, and we have some additional photographs of younger children to add to the attraction. Thank you, Sage Publications! We have paid more attention to younger children and taken out some material that was possibly beyond the age range of the book (3 to 8 years). We have a new chapter 7 on Understanding Place Value, which includes discussion of column calculation methods for addition and subtraction, always emphasising the importance of learning with understanding, of course. We have rewritten extensively the final chapter (Understanding Problem Solving and Reasoning in Mathematics) to make it more accessible to teachers in this age range.

Of all the books I have written or co-authored, I still think this is my favourite – and the new edition even more so. So I'm pleased that Sage has found these three wonderful quotes to put on the back cover:

'This book was a delight to read. The mathematical content is excellent and the approach to explaining complex concepts is exceptionally good!' -Dr Jennifer Way, University of Sydney

'I'm a really big fan of this book: it is the single most influential text in my experience of working with primary maths teachers in the last 12 years' - Andy Tynemouth, Every Child Counts National Adviser, Edge Hill University

'Every teacher of maths should read this book! It helped me realize why some children are struggling with doing simple word problems' - Amazon reader review