When I started in teaching, there were sets of textbooks on the shelves; there were dog-eared, scribbled on and torn copies of Alpha and Beta maths. I’ll leave you to guess which were more challenging. The school had introduced Fletcher maths, created by Harold Fletcher. The appropriate teacher’s book for every level, to go with the set of textbooks, was available. The head teacher at the time was also a County advisory teacher for maths, so had good background and also very good access to some of the County inspectors who regularly visited.
A significant aspect of these visits, and the accompanying discussions, was that they would rarely refer to the textbooks, which were a distillation of the underlying concepts that were fully explored within the teacher guides. It was the underlying thinking that mattered, not just getting through the books.
The phenomenon of “getting the books finished” can become a driving force when a textbook is used as the core of a school’s approach to any subject. It means that ensuring coverage trumps children’s understanding, leaving some children with gaps in their learning that they carry into the next phase. This could also now be argued with year based curricula such as we currently have in England.
Mathematics has long been decried as a national need for improvement, and, over the past twenty years, since the National Strategies, has, in many ways become ever narrower in outlook, to become arithmetic, rather than mathematics. Number facility is critical, but the practical aspects of learning can enable the practice and application of number in meaningful ways, leading to broader ideas such as data handling from a real life context.
Reference to the teacher guides for any maths scheme offers insights into the underpinning principles of the mathematics, provides the appropriate vocabulary and avoids the teacher regressing to inappropriate language interpretations which may have derived from their own childhood. Kind teachers can offer metaphors which move a child away from the maths. I can remember as a child being told about the “bottle on the step”, to deal with addition and bridging ten.
Mathematics, to some extent, exists for people to make sense of the need to count things and probably started as some kind of one to one matching system, as a tally or knotting system, allowing herders to keep a track of their flock numbers. Other needs pushed the numbers to a point where groupings, or sets of numbers supported a system of values, based on ten-ness. The incorporation of a zero, from, we are told, Arabic scholars, allowed place values to include the empty set.
Mathematics, as it largely relates to the real world, can be captured within alternative objects, matching one to one, thus allowing groupings for different purposes; 2s,3s,4s,10s, etc. Zoltan Dienes created a multibase system that underpinned the maths curriculum of my second Primary School, again with a County maths advisor as head.
Working from concrete apparatus through drawn models to mental imagery, the children developed a very clear facility with numbers, based on elements such as the function machine idea; capturing the idea of “something happening” to the original number, having passed into the machine. Exploring more challenging ideas with multi-function machines, allowed children to analyse what was going on, so that a 15 times table machine would have three elements, multiply by ten, multiply by five, add the two together. Mental maths was supported by such imagery, but those who needed it could use drawn or concrete imagery. The analytical thinking process was what was important. This was always shared, in a kind of “debrief, or sharing” (I’m sorry, I don’t like the word plenary).
I have to say that this was 1979-82.
Making sense of the development of mathematical thought is an essential precursor to being able to teach it effectively. Counting is not, of itself mathematical. It could be seen as part of mathematical poetry, words that repeat every time one restarts the counting process. Young children can learn the poetry without understanding the concepts. More than, less than, conservation of number, commutative laws, etc, all play a part in the development of understanding.
Young children using multilink for counting need to be moved to more formal systems to be able to explore fully what happens with ten-ness, hundred-ness and so on. Over-reliance on the unitary value of multilink can hinder later thought. Using metre sticks for measuring enables the incorporation of larger numbers, as a carpet of 2 metres 45 centimetres is 245 centimetres long. Children get used to bigger numbers naturally and the “playing with” or rehearsal with numbers is an important means to embed them more firmly.
Mathematics, for most purposes, is a practical skill and should retain that base.
It does become abstract and can incorporate aspects of algebra early, with empty set challenges; () + 6=14, which could be presented as x+6=14, what’s x?
My understanding of Shanghai maths, currently being advocated, is that concrete and mental modelling are central to the process of learning maths, as I have described earlier.
Real understanding, by teachers, should be the baseline for development, so that challenges developed for children are progressive in nature, rather than stand-alone activities, over reasonable timescales, to enable embedding and security. There needs to be reflection of the links between concepts, to be made overt, so that children are not continually learning new things in isolation. The growth and development of the subject should be fundamental elements, layering and revisiting, using earlier experience as the platform for progress, through judicious use of concrete and drawn models. Too often, this style of approach is sidelined for pure number, when understanding, at least of a significant proportion of the class, might be insecure.
So, rather than spend money on textbooks, I’d be an advocate of well-selected apparatus purchase, but with an investment of time in really understanding how the whole is put together and how that impacts on particular year groups, thus creating a holistic approach to the subject development. This should be repeated regularly, captured in some kind of school handbook and new teachers well-mentored into god ways of working.
Avoid the sparkly worksheets, ban the photocopying of 30 sheets and get children talking maths. Talking allows analysis or diagnosis of need. Heads down over a sheet won’t, unless it is specific to the child.
I can imagine a closure day, or a series of twilights, in the school hall, with apparatus and year-group grids, exploring the use of apparatus in each year group, “passing the baton” to the next year group to explain how they use it. Understanding the whole learning journey also underpins support for children whose learning may be less secure or less complete, ie, those who may have specific learning issues or SEN. This is particularly important in “receiving” schools, who need to have a good awareness of what has gone before, to avoid under-expectation, or to deal with their own over-expectation, leading to “deficit teaching”.
Talk maths, model maths, create lasting images. It’s all in the mind, not in the textbook.
Thinking is free; quality thinking time may need to be bought or allocated. Teaching is a team game. Let's focus on team development, not just the "hero(ine) innovator" maths leader.