If modelling thinking is a key element of teaching, why do we appear to remove concrete apparatus when the mathematical concepts start to get more difficult?

Based on an earlier blog: https://chrischiversthinks.weebly.com/blog-thinking-aloud/talking-maths

It would be interesting to know when the concrete apparatus is withdrawn from the teacher explanation repertoire, as this can be seen as only useful for SEN children, yet, used effectively, can enable even more able learners to make connections through very clear visual manipulation. This can be demonstrated through the use of Dienes base 10 material to explore place value and four rules with decimals to three places, using the 1000 block to represent 1.

How do you know what a child is thinking unless you ask them directly to explain something?

We have become used to Talk for Writing, so why not Talk for Maths? If teachers and children engage in learning dialogue, the teacher can get a better view of how the children are thinking and the learners might become more secure in their willingness to have a go, especially when facing novel situations. We also talk of it being ok to make mistakes, especially in the context of Growth Mindset thinking. I would suggest that an openness to dialogue underpins GM, in that a child should be able to share insecurities and to be able to talk through a resolution. Learning to think and talk is an important stage in being able to do so internally, from the scaffolds developed through discussion and manipulation.

The early days of my teaching career were in a school that focused its approach on the work of two key figures in mathematics; it helped that the head was a County adviser for maths, so we also benefitted from regular visits by his colleagues in the inspectorate.

There were two key elements highlighted, logic and modelling mathematical thinking supported by continuous use of structured materials. The work of Zoltan Dienes was central, embodied in the structured approach created by Harold Fletcher, whose workbooks were the spine for mathematical activities by the children. For both, we were given the key background texts to read and understand. In this way, we avoided falling into the trap of just doing the activity booklets, as both the teacher guides and the senior staff accentuated the central place of concrete apparatus. This, in itself, was accentuated through staff training as a group or 1:1 coaching to need.

From https://www.stem.org.uk/elibrary/resource/30000 an extract of Fletcher’s background.

Harold Fletcher was seen as an outstandingly gifted teacher and educationalist. While he was always a firm believer in children being able to calculate accurately, he found from his own teaching that they could achieve remarkable results in other aspects of mathematics. Harold Fletcher considered the mathematics he wanted children to learn under six strands:

Number Pattern Shape Pictorial Representation Measurement Algebraic Relations.

With the help of a team of experienced teachers and educationalists, Harold Fletcher wove these strands into a teaching sequence which was called Mathematics for Schools. Examples of classroom activities are used to describe the mathematics, complete with teacher dialogue, diagrams and outcomes from recording.

Each element of number, addition, subtraction, division and multiplication along with place value was developed showing the use of concrete materials and styles of notation (many of which would be seen later in the Framework for Teaching Mathematics (NNS; National Numeracy Strategy).

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Shape begins with an introduction to solid shapes before bringing in 2 dimensional or plane shapes. This is followed with measuring, area, capacity and volume before concluding with symmetry and tessellations. As with all aspects of the series it was stressed that concrete materials should still be used.

Pictorial Representation focused on students, from an early age being able to collect information, record it in pictures and most of all, think about it and use it for further number practice. The foundations for graphs were introduced before dealing with them further in Algebraic relationships. A final section on “How can I help my child?” contained some do’s and don’ts. A pdf of a parent guide is available from the STEM site above.

The second key character in my mathematical education as a teacher was Zoltan Pal Dienes (Pal anglicized to Paul). Looking up some detail, I came across his relatively recent obituary.

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DIENES, Dr. Zoltan Paul Obituary from http://www.zoltandienes.com/obituary/

Age 97, of Wolfville, Nova Scotia, passed away peacefully on January 11, 2014. Zoltan Dienes, internationally renowned mathematician and educator, was both a public figure and a much loved family man.

Zoltan was born in the Austro-Hungarian Empire in 1916, son of Paul and Valeria Dienes. His early years were spent in Hungary, Austria, Italy and France. He always had a fascination with mathematics, even hiding behind a curtain to hear his older brother’s maths lesson, for which he was deemed too young!

At 15 he moved to England. He received his Ph.D. from the University of London in 1939. Zoltan understood the art and aesthetics of mathematics and his passion was to share this with teachers and children alike.

He was fascinated by the difficulties many people had in learning mathematics and wanted others to see the beauty of it as he did. Consequently, he completed an additional degree in psychology in order to better understand thinking processes. He became known for his work in the psychology of mathematics education from which he created the new field of psychomathematics.

Referred to as a “maverick mathematician”, Zoltan introduced revolutionary ideas of learning complex mathematical concepts in fun ways such as games and dance, so that children were often unaware that they were learning mathematics – they were having a wonderful, exciting, creative time and longing for more. He invented the Dienes Multibase Arithmetic Blocks and many other games and materials that embodied mathematical concepts.
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According to a Montana Mathematics Enthusiast monograph from 2007, “The name of Zoltan P. Dienes stands with those of Jean Piaget and Jerome Bruner as a legendary figure whose theories of learning have left a lasting impression on the field of mathematics education…

Dienes’ notion of embodied knowledge presaged other cognitive scientists who eventually came to recognize the importance of embodied knowledge and situated cognition – where knowledge and abilities are organized around experience as much as they are organized around abstractions. Dienes was an early pioneer in what was later to be called sociocultural perspectives and democratization of learning.”

I had a wry smile when I realised that his initials are ZPD, which is immortalised in the work of Vygotsky as the Zone of Proximal Development, something, I am sure that Dienes would have appreciated.

Conservation of number became a shot topic of conversation on social media during the week. It is, without a doubt an underlying concept in the learning of mathematics, akin to chunking of information to make subsequent thinking and manipulation easier.

Definition
Conservation of numbers means that a person is able to understand that the number of objects remains the same even when rearranged.
What is conservation of number?

• Conservation of number - the logical thinking ability to recognise that the numerical value of an object remains invariant with physical rearrangement - is a fundamental "cognitive milestone" during children's development (Crawford, 2008 p. 1). 
  
  
• The concept of conservation was developed by Jean Piaget during the mid-1900s, who claimed it as "concrete operational" and, therefore, "unattainable" until children are of 7 or 8 years old (Halford & Boyle, 1985, p. 165). 

It is interesting visiting schools and classrooms, watching many numeracy lessons. It is often clear that children are regressing to counting from one, which suggests that they have not reached the conservation stage, even when dealing with relatively small numbers. This might be down to lack of modelling, therefore expectation, with high adult oversight and interaction.

Some of the materials being use for modelling may be less helpful, in that they might encourage children to start counting from one, for security.

Multilink or Unifix blocks are common in early classrooms. Where the mathematics takes children beyond tenness, breaking the chain into ten rods can be useful to accentuate that concept. It is heartbreaking to watch a child count, then have to restart the count because they have been interrupted. Making rods of ten would allow for interruptions and a means of continuity.

For this reason, I still have a preference for Dienes base ten equipment, as it allows early access to models of exchange, creating tens then hundreds. It accentuates place value and, using the function machine conceptualisation allows all four rules of number to be modelled effectively. With a visualiser attached to a class IWB, the modelling can be done on a large scale, enabling more to access. I recognise that in the days before such technology, there were visual limitations to modelling to large groups.

From Maths No Problem, the following accent on concrete apparatus seems to fit with this approach.

Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the Singapore method of teaching maths for mastery.

Recently working with ITE trainees, in the conversation the idea of conceptualisation was raised. Using the simple example of 2+2=4, the trainees were challenged to explore the underlying necessary concepts to have a full grasp of the challenge. Twoness, fourness, addition (including synonyms) and equality, or balance, linked to balanced equations, eg 2+2=3+1. It was an eye opener to some.


​Race to the flat
A very simple activity that can be very effective in supporting rapid calculation could be called race to or from the flat.

As long as you have Dienes base 10 materials and dice, this can be developed to cater for a variety of needs.
The rules of each game are simply described.
·         Decide whether it’s a race to or from the flat (100 square). Decide whether, when the dice are thrown, the numbers are added together (any number of dice) or multiplied (two or three dice?).
·         Dienes materials available to each group, plus dice appropriate to the needs of the group.
·         Each child takes turns to throw the dice and calculate the sum or product.
·         This amount is then taken from the general pile and placed in front of the child. The calculation can be recorded eg 3+4=7. This can provide a second layer of checking.
·         If playing race from the flat, the child starts with ten ten rods, then takes an appropriate amount from these.
·         Subsequent rounds see pieces added to the child’s collection; recorded as needed, eg round 2, 5+2=7 (7+7=14; the teacher should see one ten and four ones)
·         The first child to or from the flat is the winner.
Altering the number of dice alters the challenge.

An extension could be a race to the block (1000 cube), or from the block, each child starts with ten 100 squares. If multi-sided dice, or different numbers of dice are available, the challenge alters yet again.

I came across some notes from some years ago, where I sought to put together examples from Dienes to enable colleagues to utilise the system to support their maths teaching.

Something from my file...
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A starter for an activity that could be developed in many directions, depending on the needs of the children. Shape and pattern in one activity?

As long as you have Dienes base 10 materials and dice, this can be developed to cater for a variety of needs.
The rules of each game are simply described.
·         Decide whether it’s a race to or from the flat (100 square). Decide whether, when the dice are thrown, the numbers are added together (any number of dice) or multiplied (two or three dice?).
·         Dienes materials available to each group, plus dice appropriate to the needs of the group.
·         Each child takes turns to throw the dice and calculate the sum or product.
·         This amount is then taken from the general pile and placed in front of the child. The calculation can be recorded eg 3+4=7. This can provide a second layer of checking.
·         If playing race from the flat, the child starts with ten ten rods, then takes an appropriate amount from these.
·         Subsequent rounds see pieces added to the child’s collection; recorded as needed, eg round 2, 5+2=7 (7+7=14; the teacher should see one ten and four ones)
·         The first child to or from the flat is the winner.
Altering the number of dice alters the challenge.

An extension could be a race to the block (1000 cube), or from the block, each child starts with ten 100 squares. If multi-sided dice are available, the challenge alters yet again.

With original words by Tony Hatch, Messing about on the River was a song on weekend children’s radio in the early 1960s.

As a child growing up in Brisbane, Qld, and Torbay, boats were a regular feature of life. Being in or on the water was a particular pleasure of growing up, with one darker incident, when a friend and I were the last people to see a boat loaded with six men and several sacks of cockles, with so little draught that while we were still out rowing, the police launch drew alongside and asked us if we’d seen them, as they hadn’t got back as planned. They had shipped water and all six had drowned when their boots, filled with water, had acted as anchors. It led to my first and only appearance in a Coroner’s court.

A tweet today talking of the oral tradition and nursery rhymes, reminded me of an earlier blog, but also made me think a little further about examples from my teaching.

As a teacher of a year 3 class around 1984, the topic choice was water, which provided the science, with exploration of floating and sinking, density exploration, evaporation and the water cycle, especially on showery days, siphons and pumps. Rafts enabled exploration of area and volume, linking science and maths. Rivers underpinned the geography, a visit to the Victory for some history and, for a short period, the song was the basis for dictionary and reference book research.

In the days before mass internet availability, the use of non-fiction books, using the contents and index to seek out information that could then become a general class resource, eg within displays was a common feature. Any parent who was associated with the navy, Royal or merchant, or a sailor or boater might be asked to visit to provide a personal talk.

Occasionally, this developed into an “alphabet of…” whatever was the current topic, creating a glossary of useful terms.

The song gave the focus, with specific words being identified as worthy of exploration. Ultimately, the activity also developed in-class thesaurus-style collections of associated words. By becoming the active explorers, children then often went home and found out more for themselves. The song became the vehicle for broader language development, but also, by being learned by heart for a performance in their assembly, helped with memory.

Oracy is a current buzz word. Like many others, it seems to mean different things to different people. To me, it means giving children something of quality to talk about, in small or larger groups, with the purpose of finding a solution to a problem, or working out how they will tackle a challenge. It’s rarely as formal as a debate, but might become such in specific circumstances. The confidence to interact with peers, to me, is more important than performance to a wider audience, as that’s how we live. Few of us have a soap box upon which to stand or a lectern to hide behind.

Learning to interact verbally is a life skill. A language rich environment encourages that.
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For those of you who don’t know the song, here’s a link. I'm hoping to resurrect my interest in the water and water activities when retirement beckons... who knows, I may even be tempted to accompany this with singing! 

There’s always a need to fill in a few minutes here and there, even within a well-oiled organisation. You’ve finished and cleared up from a lesson, in readiness for assembly, when a note is circulated to hold on for five minutes while something in the hall is sorted.

You have 30plus children in a snake, ready to walk to assembly and time on your hands.

Poetry can be a great filler. I always had a set of poetry books on the shelf, to share regularly, and could then choose as few or as many as needed to fill in the available time. It’s especially useful if the children know many of the poems my heart, so can join in with the telling, as a rehearsal activity. Children like silly poetry.
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They also like silly songs, so a collection of these would be developed through the year, so that, with a few minutes and a child chosen, the line could become an impromptu choir, enjoying the feeling of performing together.

Blank playing cards offered a wide range of potential uses.

The numbers 1-100 written on separate cards allowed two or three to be selected at random, with the chosen numbers discussed by the children. This allowed discussion of larger/smaller, or greater/lesser, place values, ordering a set of numbers according to attributes. It could also become random mental maths challenges with addition, subtraction potential.

Dice come in different number combinations. These can be thrown to create random maths problems.
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I’ve even used the Dienes base 10 material, challenging children to find specific values; 36= 3 tens and 6 ones.

Blank cards can become flash cards, with particular words to be shared regularly, especially after their phonic components have been covered.

This could be extended to broadening vocabulary needs, where a word selected at random is created into a sentence, and improved around the class. This could be subject specific vocabulary, with the need to incorporate other knowledge into the sentence.

I also had three piles of cards, divided into people, places and things. Children would take one card from each pile, then create a story to link the three elements within the time available. This style of activity was occasionally developed as a writing stimulus, for longer consideration.

Images. A classroom collection of images, as paper copies, or through the IWB becomes a talking point, starting from description of what is seen, through speculation about what happened before and after the image was taken, or what was happening outside the frame of the image.

Five minute slots can be very important for rehearsal activities. It is far better to have ideas as a back-up plan, as five minutes can sometimes extend and a line of children standing around can get very restless, developing into a behaviour management issue.

My advice is always to be prepared. Have something up your sleeve, especially if you can do magic tricks!

You might like to look at one minute data ideas or five senses starter activities

​Working with an infant class, I had visited our local materials bank and come back with a large roll of card. Incorporated within another art activity, every child did hand print handspans along the card. This became our non-standard tape measure, which we used to measure (approximately) the circumference of trees in the school grounds and width of the canopy, with an extension of measuring the playground. It became clear that additional organisation was needed, so the roll was divided into sections of ten handspans. The children were able then to use the available resource and some tallying to measure the width and length of the playground using the strips of ten handspans.

The strips also doubled as the border for a display, so had multiple purpose.

We did an experiment with each child using their feet to measure the length and width of the classroom. The results were counted and tallied and, surprise, there was a range of result, which lent themselves to a variety of recording methods; arrow charts, lists, bar charts…

I made up a story about a king who asked a carpenter to make a boat that was one hundred feet long. The carpenter worked diligently, and created the most magnificent boat ever seen. When the king came back he took one look at the boat, admired it’s elegance and then put the carpenter in jail for creating a boat that was too short…

The children had to discuss why the king might be angry. The need for agreed measures was brought up and they all agreed that the king should have left a pattern of his own foot as the unit of measurement. A wider discussion explored the basis for standard measures across everyone and the activities were repeated using metre sticks or tape measures.

Measures were always a significant part of much planning; most Primary topics effectively lend themselves, at some point to being able to measure something, in some form, using standard or non-standard measures. It allows working mathematically out of doors, with the results able to be collated and recorded in a variety of forms appropriate to the needs of the age group, introducing young children to data forms early.
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So, as a rule of thumb, I’d sometimes fill in odd moments with a bit of measuring. It’s surprising how much pleasure young children get by finding that their bodies are natural measures.

It ain’t what you know, but the way that you use it…

It’s been a bit of a week, probably a good one to go out and buy a hard hat, if you spend any time on Twitter. It wasn’t just raining, it was, on occasion, literally pouring. Bile ducts were seemingly emptied on the heads of a few writers willing to proffer views with were opposite to others; as if people can’t see things in a different light. If it wasn’t knowledge, which it largely was, it’s overtaken by some element of English teaching, but that invariably comes back to children’s lack of knowledge that can be utilised within their oral or written efforts or in their ability to decode the written word.

The two “camps” could be described as those who think they can impart knowledge by sharing a specific body of knowledge, within their classroom teaching and those who seek to develop this knowledge through a variety of linking experiences, including the spoken word.
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This can sometimes appear to be the divide between (some) Primary and Secondary practitioners, with the extended argument that Primary learning is all discovery and play. That the approach is sometimes different, I wouldn’t want to argue, it can become a moot point, but, in reality, it’s likely that there is more convergence than divergence. Whereas young children “play with ideas” through active engagement and sometimes concrete examples, older children, hopefully, are more able to “play with ideas”, so have greater insights from their developed vocabularies.
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There was a significant announcement yesterday, 12th July 2016, where the Schools’ Minister, Nick Gibb, announced a £40million budget to embed “Shanghai Maths” into 8000 British Primary schools. This was for textbooks and, we assume, staff training. That this works out at £5000 over apparently four years, or £1250 a year per school, doesn’t seem like a great sum, even on my simple calculations, with training cover somewhere around £170-200 a day.

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.

Number is, and always has been, an essential aspect of mathematical education. Almost everything can be reduced to numbers in some form, if only counting. "How many ...?" is a question that is regularly asked by children or their parents, with the inevitable counting that ensues. The (cardinal) counting system, based as it is on the simple concept of +1, is a part of the "poetry" of growing up. Parents count to and with their children, as simple counting, or within nursery rhymes or songs; one, two, three, four, five, once I caught a fish alive.

Sometimes, it is extended to the ordinal form, as in first, second, third.

​However, counting, although a baseline element of number, is not the be all and end all of maths, but it is one aspect to which children regress, if they are less secure in developmental stages.

​On entering my first Primary classroom, the maths resources were based on matching and grouping cards, similar to the cotton reel picture below, with many different variations on real objects and clip art pictures, which then had to be matched, as 1:1 correspondence with some kind of counting material, as in picture 2, with the reels removed.

​This exploration and "replacement" of one element with another, is an essential step on the road to more abstract interpretation and manipulation of ideas that have their base in the real world.

​Before moving on, try to estimate how many reels there were, either from the picture or the counters. I wonder how many readers will have automatically counted them?

Grouping the counters into groups of two, three, four, etc allows insights into the broader properties of the number being explored. In this case, 7 groups of 2, 4 groups of 3 and 2 extra, 3 groups of 4 and 2 extra. The properties demonstrate an equality which is fundamental to conservation of number.

Eventually, grouping into tens was reached, beginning then to demonstrate the relative values of a collection of ten and the unitary elements. Once you can "read" the image, the group of ten equates to the number 1 in 10, as long as the learner understands the need for the 0 to signify an "empty set". So far, we have explored 1:1 correspondence through matching, grouping and set theories.

Having established a baseline understanding, the learner has to accommodate the idea that numbers can be manipulated, through what we call the "four rules (or functions in the language of Zoltan Dienes) of number", addition, subtraction, multiplication and division.

​Essentially the learner has to understand that they are effecting a change of state, where the number that you started with is acted upon by an "instruction".

​Addition, already encountered in the counting system, essentially entails the understanding that the original number is being enlarged by a specific amount. this can entail counting on from the "conserved" number, not from 1. If a child regresses to counting from 1 they have not secured conservation of number. 

​Linking the counting element to some kind of number based visual system, like Dienes or Numicon, adds another visual, but this can lead to visual representation, diagrammatic modelling, which can then be transposed to the number system.

​"Greater than, less than, difference between", are all underlying concepts that are explored early in a child's maths experience. Difference between does not need to imply subtraction, but often is presented as such, which can introduce unnecessary complications and potentially remove effective methodologies.   
 

I have not mentioned number lines or 100 squares, or any other systematised supports. These need to be introduced with care, if they are not to add any layers of confusion, that can then cause misconceptions that have to be unpicked. In the attempt to "simplify", especially for struggling learners, it is all too easy to add complexities that confuse, rather than support. 

​In my early classrooms, where "maths resources" may have been picked up during a walk at the beach or the woods, a trip to a charity shop, or some other means of sourcing artefacts or counting supports, improvisation was needed.

​Any form of counting material enabled exploration of arrays, underpinning the commutative law , that 3*4=4*3

​Equally, we sometimes explored square numbers as in the picture below.

​The principle that maths is everywhere is an important one. It should not exist in the purity of the maths lesson, but needs to demonstrate that what is learned in maths has application across many areas of life.

Linked blogs
​Talking maths
Maths everywhere
Show maths, talk maths, draw maths, image maths.
Investigating mathematically
More maths Activities
Quick (one minute) data
The answer is twelve?
Story maths?

Move it, draw it, think it, talk it……
Start small and grow thinking.
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A couple of months ago, I was asked to lead a training session on feedback in mathematics. I spent some time reflecting on the practicalities of this before accepting, as I could see a number of potential pitfalls and blocks, dependent on the school and individual thinking and working methods.

In the end, I decided to take as a central theme talking maths. This would allow me to explore presentational ideas as well as interactive elements of a lesson.

All thinking about teaching and learning comes back to the learner and the teacher understanding or their needs at a particular point in time, with regard to the specifics of the next steps in teaching. This knowledge should build on their previous experiences, with an overview of prior attainment, both of which will determine the means of sharing the new ideas, links with the prior learning, but also to consider the need to explain and model the new information in the light of individual needs. This modelling might be through concrete apparatus, visual diagrammatic representation or oral modelling, if the learners have secure internal models, which they can manipulate.

It would be interesting to know when the concrete apparatus is withdrawn from the teacher explanation repertoire, as this can be seen as only useful for SEN children, yet, used effectively, can enable even more able learners to make connections through very clear visual manipulation. This was made clear in the training session when I demonstrated the use of Dienes base 10 material to explore place value and four rules with decimals.

How do you know what a child is thinking unless you ask them directly to explain something?

We have become used to Talk for Writing, so why not Talk for Maths? If teachers and children engage in learning dialogue, the teacher can get a better view of how the children are thinking and the learners might become more secure in their willingness to have a go, especially when facing novel situations. We also talk of it being ok to make mistakes, especially in the context of Growth Mindset thinking. I would suggest that an openness to dialogue underpins GM, in that a child should be able to share insecurities and to be able to talk through a resolution. Learning to think and talk is an important stage in being able to do so internally, from the scaffolds developed through discussion and manipulation.

Language is key. Using the correct vocabulary and ensuring that children do so, underpins a mutual understanding, and may require interpretation and linkages to a broad range of synonymous language, to ensure all understand. In my opinion, it is fine for a learner to ask for a reminder. Asking supports Teacher Assessment, in that it might demonstrate a level of insecurity, which needs to be addressed to avoid this getting greater.

Asking a child to explain the steps they would take to solve an equation offers an opportunity for writing instructions, or reportage, but also links with a “Show your working” approach, which I would also advocate. Either way, I’d be looking to have as much information as possible available to review outcomes in the round. It is very easy to see maths as producing right or wrong answers.

Talking the steps and showing your working, with apparatus, written models and written methods, would, for me underpin any investigative approach to understanding a child who may be expressing difficulty.

This should be a teacher level activity, so that any remediation needed, perhaps in the hands of a Teaching Assistant, can be focused to the real needs, rather than assumption. The mathematical thinking of the TA needs to be considered, so that “short cuts” and alternative methods are not deployed to make it seem as if the child is getting it right, when they have underlying issues.

Children should be supported in their confidence throughout, encouraging effort, exploring alternative scaffolds and materials as needed, removing these when they are beginning to show confidence. It is also important to demonstrate the links between the scaffolds, eg number lines, number squares, Numicon, Dienes, so that they can select if need arises.
 
Using the analogy of teacher as storyteller, it is important that children are told the story in such a way that they can see the storyline and the developing detail, as it gets progressively harder. Activity should be accompanied by modelling, in a form that supports each learner’s needs. Articulation, from both teacher and learners should be a high priority, as this provides the insights that guide teacher decisions. Just marking the books can often give false information.

Keep talking mathematically, across all subjects, so that it is clear to learners where it can be used and applied. They must learn that maths is all around us, from an early age. Everything can be counted or measured in some form.

Linked posts
Maths everywhere
Show maths, talk maths, draw maths, image maths.
Investigating mathematically
More maths Activities
Quick (one minute) data
The answer is twelve?
Story maths?
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