Do you teach mathematics? If so, do you know the difference between perceptual subitising and conceptual subitising? Do you know what the cardinal principle is? Do you know how five-frames and ten-frames can play a role in developing children’s sense of number? If any of these leave you stumped, then you are not alone. From my experience, too few maths teachers at KS2, KS3 and beyond understand the mathematics journey that their students have been on and the potential foundational mis-steps that might be causing their students difficulty. Until fairly recently, I knew next to nothing about early mathematics. I’m far from an expert now, but the knowledge that I have gained on the subject has come a decade later than it should have done and has transformed how I view the teaching of mathematics. If, like me, you would like to learn more about this crucial area, then here is my guide on where to begin, which tries to take into consideration the amount of time that you have to dedicate to it:
If you effectively have no time to dedicate to this goal at present, then get the ball rolling by following these people on Twitter:
Mr Westacott combines infectious enthusiasm with expertise and insight. Here he is discussing the use of manipulatives with @mrbartonmaths: https://www.youtube.com/watch?v=HwQYVwbOgdw&list=PL7BJ-1MkmUZ9A6m4qYXgISzZy63CQT4sh
Dr Williams is a thoughtful and incisive advocate of research-informed practice in EYFS, and her patience in discussing early mathematics with neophytes like me is very much appreciated. Here she is discussing early maths, again with @mrbartonmaths:
This gent knows his stuff. Follow him and then bookmark any tweet he makes on the subject of mathematics.
If you have 2-3 hours only, explore the Learning Trajectories website and read Making Numbers:
This website is arguably the quickest way for a novice of early mathematics learning to become better informed. The Learning Trajectories approach is to attempt to specify a progression that learning can follow for various areas of early mathematics, from subitising to spatial visualisation. For each area, there are several levels with each demonstrated through brief videos and learning activities. If I ran a primary teacher training course, I would give every trainee and hour or two, at least, to explore this website. It is a goldmine.
Simply sign up for free, set up a class (it doesn’t need to be populated with students, just given a name) and away you go:
Making Numbers by Rose Griffiths, Jenni Back and Sue Gifford
I can’t imagine a more welcoming, accessible introduction to early maths pedagogy than Making Numbers. Full of simply expressed ideas that belie the depth and utility of the underlying concepts, this book is a relatively quick read, but one that you will return to repeatedly, especially if you teach maths in reception or Key Stage 1. The only drawback to this book is its cost. I’d recommend persuading your maths coordinator to fork out for it as this book would be a useful addition to any CPD library.
If you have 6 hours, also read this book:
Understanding Mathematics for Young Children by Derek Haylock and Anne Cockburn
This book is the perfect place to start. It is accessible and informative and provides one framework for getting to grips with the idea of ‘understanding’ in mathematics. It discusses early number, operations, the principles of arithmetic, shape & space and problem-solving in highly practical ways, but also manages to grapple with what it means to think mathematically.
Highlight: within the Understanding Shape and Space chapter, there is an analysis of this topic that progressively moves through subtle changes to shapes, illuminating the crucial mathematical thinking in finding differences and similarities. It is an implicit lesson in the use of variation and an insightful look at what is meant by ‘equivalence’ in mathematics.
If you have 10 hours, also read these two books:
Teaching and Learning Early Number by various authors and edited by Ian Thompson
A collection of short essay-like chapters by various researchers into early maths, this book unpicks many of the complexities of children’s early ideas. Each chapter can be read as a standalone exploration of a given topic, which makes it an easy read, but it can feel a little disjointed. Regardless, it’s well worth your time.
Highlight: Thompson’s chapter on mental calculations, in particular the extra complexity hidden in some supposedly simpler methods of calculation, discusses why getting stuck on certain strategies of addition and subtraction can lead to unnecessary struggles over the longer term.
Teaching Mathematics 3-5 by Sue Gifford
While the books mentioned above provide a useful route to understanding early mathematics for all primary and secondary educators, Teaching Mathematics 3-5 is probably the most useful for an educator working with children across the EYFS age range. Gifford presents a research-informed view of early mathematics, addressing the need for a holistic view of children’s learning that respects the social, emotional and physical aspects of the journey to understanding. She emphasises the need for adult-initiated mathematics learning (contrasting this with adult-led activities) within stimulating, thoughtfully constructed learning environments. In addition, Teaching Mathematics 3-5 is full of practical suggestions and snippets from real interactions with children. Often these snippets are of children’s misconceptions, making the book a welcoming read for those new to the profession and one that will resonate with more experienced teachers.
Highlight: The brief section on playfulness and humour in mathematics learning stands out as a subject that is too rarely discussed on Edutwitter or in other texts, and yet is a valuable component of expert teaching.
If you have 20 hours, also read these two books:
Growing Mathematical Minds by Jennifer S McCray, Jie-Qi Chen and Janet Eisenbard Sorkin
This is an attempt to join the findings of early mathematics research to the practicalities of classroom teaching by giving teachers the chance to enter conversation with researchers. While I don’t agree with every interpretation made within the book, it is a worthwhile exploration of how research can impact real settings. I particularly appreciated Siegler’s ‘overlapping waves’ model for the way that children use different calculation strategies under different circumstances helps tie the complexity of real learning to the occasionally simplified categories in academic research.
Hands On, Minds On by Claire E Cameron.
This book details the research on executive function, motor skills and spatial skills and how they relate to early learning, the last of these in particular being implicated as having a relationship with later mathematics learning. It’s a fascinating look into the foundations of all school learning.
Visible Maths by Pete Mattock.
Visible Maths is not directly related to early mathematics; in fact, much of its content is most useful for secondary teachers. Nonetheless, its explanation of how the use of manipulatives and pictures can enhance learning is very useful for teachers of early mathematics.
If you have more than 20 hours, also consider these books. (I am currently reading these, but they seem well worth the effort):
Learning and Teaching Early Maths – The learning Trajectories Approach byJulie Sarama and Douglas H Clements.
Early Childhood Mathematics Education Research: Learning Trajectories for Young Children byJulie Sarama and Douglas H Clements.
These are rather expensive, but come highly recommended from people who know considerably more than I on this subject. Both are related to the Learning Trajectories website that is discussed above. The first of the two books is more practical, while the second is a description of the research upon which the Learning Trajectories approach is based.
If you’re interested in mastery approaches to mathematics, I’d highly recommend Mastery in Primary Mathematics by Tom Garry.
And that’s about it. I hope you get as much from learning about early mathematics as I have.