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?
Something from my file...
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?
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 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.
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!
When the weather is fine then you know it's a sign
For messing about on the river.
If you take my advice there's nothing so nice
As messing about on the river.
There are long boats and short boats and all kinds of craft,
And cruisers and keel boats and some with no draught.
So take off your coat and hop in a boat
Go messing about on the river.
There are boats made from kits that reach you in bits
For messing about on the river.
Or you might want to skull in a glass-fibred hull.
Just messing about on the river.
There are tillers and rudders and anchors and cleats,
And ropes that are sometimes referred to as sheets.
With the wind in your face there's no finer place,
Than messing about on the river.
There are skippers and mates and rowing club eights
Just messing about on the river.
There are pontoons and trots and all sorts of knots
For messing about on the river.
With inboards and outboards and dinghies you sail.
The first thing you learn is the right way to bail.
In a one-seat canoe you're the skipper and crew,
Just messing about on the river.
There are bridges and locks and moorings and docks
When messing about on the river.
There's a whirlpool and weir that you mustn't go near
When messing about on the river.
There are backwater places all hidden from view,
And quaint little islands just awaiting for you.
So I'll leave you right now to cast off your bow,
Go messing about on the river.
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.
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.
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
Meaning; A means of estimation made according to a rough and ready practical rule, not based on science or exact measurement.
Origin; The 'rule of thumb' has been said to derive from the belief that English law allowed a man to beat his wife with a stick so long as it is was no thicker than his thumb. In 1782, Judge Sir Francis Buller is reported as having made this legal ruling and in the following year James Gillray published a satirical cartoon attacking Buller and caricaturing him as 'Judge Thumb'.
When I was around six years old, an uncle gave me a tape measure as a bit of a plaything, to keep me out of mischief during a holiday. It was a good ploy, as I spent much time wandering around measuring everything I could literally lay my tape upon. It started a fascination with measures, which lasted.
Later, I studied the Romans and got interested in their measuring system, for example, the pes, or Roman foot, pollex or thumb and the palmus or palm, together with the cubit and gradus or step. I discovered that my thumb from knuckle to tip was approximately an inch or 2.5cm, while my hand span was 9 inches or 22cm. Counting the steps between lampposts was a short term slight obsession, to find out if it was always the same. Uncle Don has a lot to answer for…
This still is the case. It can be quite useful if out without a tape measure, to know that four handspans would equal approximately 36 inches or one yard, in old measures, or five handspans, not quite flexed would approximately equal a metre. It can be useful when considering a piece of furniture or similar, to estimate whether it would fit or not.
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.
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.
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.
Now, I don’t know about you, the reader, but, if I wanted to teach someone about, say, castles, which is/was a regular element of Primary life, to sit and talk about castles could be interesting. It’s relatively easy, within initial planning, to write a list of, say, twenty key aspects of castles that the teacher deems essential to be covered and understood within the topic. If however, half or more of the class had not been to a castle, the children may not have the means to engage with the details.
Words like drawbridge, portcullis, motte, bailey and keep might be explicable, but what about barbican and belvedere, casements and crenelation. They might have fun with the idea of cesspits and garderobes… As for the fighting people, their support and family lives, with the attendant additional vocabularies and layers of understanding, the complexity grows.
So, as a Primary teacher, wanting to interest children in such an area, it’s likely that some kind of site visit would assist, particularly if there is a local example. If this can be guided in some way, by a local expert, this can add colour to the visit and deepen the narrative. Any need to interpret what was said can be done by the teacher in follow up discussion.
Particularly in the early days, but throughout my teaching career, a display of available material would support the topic, both in picture and in book form. Later, this would also include video or DVD material to be shared, as a class, or within groups. All to provide some additional background and stimulus.
Before the internet opened up search options, the books were a key element of the reading curriculum, extracting appropriate information from using the contents list and the index, to provide answers to pre-set questions. This might be extended with a request to record three/five additional interesting items of information. The research would be shared and sometimes collated in a displayed alphabet of the topic; effectively developing our own glossary. Try http://www.castlesontheweb.com/glossary.html if you’re interested.
DT was deployed to make models, of drawbridge and portcullis “mechanisms”, using pullies. Castle models were built, dolls dressed, food prepared and cooked…
Sketches from the visits were developed into larger pieces, added to from the available imagery. Photographs were taken, developed (taking a week), then used as storyboards.
Drama situations were set up to re-enact situations and seek some kind of further understanding.
While specific elements of history were relatively easy, geography might be developed through an exploration of where people chose to position their defensive sites, but also consideration of material availability and movement, the availability of water and food.
Science might be developed through trajectory exploration of a range of objects, or material strength, including exploration of elements like lintels across openings.
Throwing things could also link with PE…
With the Normans, Portchester Castle is very close, it was also possible to look at the language that came with them, at an appropriate level of course. So we might look at cow and beef, pork and pig, mutton and sheep.
In many ways, thinking as a Primary teacher automatically seeks to incorporate the curricular range available within a specific topic, without seeking to shoehorn in ideas just to be cross curricular. However, it does demonstrate that, so far, every area covered allows language development.
Mathematics from building exploration can include shape, measurements using age appropriate forms; with year six, we made a clinometer to work out an approximate height. Setting a challenge to estimate the number of blocks used to build the castle allows for some estimation, but also calculation, to gain a rough idea.
And how were the stones cut? What was the life of a stonemason like? How did they build their castles ever higher?
Essentially, you could take any topic and take it to post graduate degree level. Some teachers will have done, in a relatively narrow field of expertise. The information shared with children has to be age appropriate, using language forms that are understandable to the children and interpreted to those who don’t have an understanding.
It is reasonable for a teacher to ask whether they know enough about the topic and to create checklists of information that they think will come in handy, as aides memoire. These then inform planning decisions. Some are calling them “knowledge organisers”. Where they are described as to be taught and then tested, with under-confident/early career colleagues can lead to that being the approach. Making a topic broader, going beyond the skeleton to put real flesh on the bones can take deviation from plans and adding value to agreed approaches. When a confident teacher is able to fully develop the learning narrative, the children engage further and, in my experience, then start bringing in aspects that they have done at home; a picture, model or some writing from books at home.
We have to accept that, as learners, children are in the process of learning.
The teacher is the leader and their guide throughout. The teacher if map creator and reader, deviating to the evident need of the group or individuals, stopping, taking stock, pressing on and adding further, with hopefully all arriving safely at the preferred destination. Some will get messy on the way, having struggled through the muddier elements.
Hopefully, even after a good picnic, which they’ll always remember as a highlight, they are hungry for more.
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.
Move it, draw it, think it, talk it……
Start small and grow thinking.
This collection of statements passed by my Twitter window recently, extracted by Prof Rob Coe of Durham Uni, from a Harvard study by David Blazar. http://cepr.harvard.edu/files/cepr/files/blazar_2015_effective_teaching_in_elementary_mathematics_eer.pdf
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.
Show maths, talk maths, draw maths, image maths.
More maths Activities
Quick (one minute) data
The answer is twelve?
Every equation tells a story
Explain how you would tackle the problem; 37+45+26
As a child at school, we were always exhorted to show our working, then we’d get credit, even if the answer was wrong. This idea of showing working is important, as, unless that is available and the only available information is an answer, the teacher has no idea what the child was thinking through the process.
One day, as an experiment in an infant class, who had been spending time writing reports, I asked them to write the story of how they tackled an addition problem. At that stage, I had no idea how it would go, but, having written reports, they were used to putting ideas into some order. With one group, they had to solve the problem talking aloud throughout, with an adult scribe.
Their writing also became a de facto script, so that they were able to rehearse to their peers what they had done. The articulation sometimes identified areas where they had missed out a stage or instruction.
Most were able to write and talk in terms of steps that they took and were able to explain to their partners what they had been thinking throughout. They had a very good context for time connectives, before they were on any curriculum.
Peer talk became, after a while, something that became a regular part of the classroom maths practice: the need to explore and explain their thinking when sorting out a problem. Working in partnership enhanced their articulation and clarified their thinking, so that, eventually, there was a marked improvement in basic arithmetic.
Showing their working became talking their working, became remembering their working.
Still browsing files. Here’s a series of maths activities that lend themselves to open ended investigation with a little tweaking, or can be used to think though specific issues within a practical base.
Investigating with cubes
Resource; tray of multilink cubes or similar.
Investigation; To explore how many different shapes can be made with 1,2,3,4,5,6 cubes?
This investigation enables discussion of reflected and rotated shapes and whether they are “allowed” as separate shapes, or whether they should be disallowed.
A table of results could be developed, to record results.
A subsidiary activity could be, having created shapes with 4 cubes, better perhaps, if each shape in single colour, to see what shapes can be made by using combinations of cubes, say 12, 20, 28.
This can give rise to drawings of different elevations, top, bottom, four sides. All can lead to mathematical discussions.
Devise a checking method to show how many squares there are in a 3 squares by 3 squares grid.
Repeat this for a square of 4 by 4 squares.
Set out results in a table.
Is there a pattern in the results? Consider square numbers.
Could you predict how many squares would be in a 5 by 5 grid, then prove it?
How many squares on a chess board?
In the same way as the squares investigation, the use of triangles might give rise to a different range of mathematical discussions.
Isometric paper allows children to draw the triangles for themselves, or they can be provided for them.
Start with one triangle side, two, three, four, etc, so that investigation is ordered, and have the children create a table for results.
After three, they can start to predict how many triangles they expect to find and to explain their thinking.
Using multilink, can you make a square based pyramid two stories high, with one cube at the top?
Three, four, five…
Enables early discussion of square numbers.
Predict the next series of layers, then prove.
Back to one…
Dividing by 2 practice.
Allow each child to choose a starter number, appropriate to their current awareness.
Rules of the task.
If it can be divided by 2, do it, if not, add 1.
33+1=34/2=17+1=18/2=9+1=10/2=5+1=6/2=3+1=4/2=2/ (2=1) repeats
If this is drawn out, linking all the numbers, a tree starts to grow.
Resource; multilink or similar linking cubes.
Discuss the attributes of a rectangle.
What’s the smallest rectangle you can make with cubes?
Make a series of rectangles of different sizes and explain their attributes.
How many different rectangles can you make using 12 cubes each? (4)
How many cubes would it take to make five different rectangles with the same number of cubes in each? (36)
Question; is a square a rectangle?
Six (60), seven (120-192), eight (120)
Back to zero “snake”
Based on the idea of function machines, each stage is an equation, with the output becoming the input into the next machine.
The start number can be any with which the child feels comfortable. NB The function machines can be hand drawn, as the task progresses, or before starting, if a specific number of equations are expected.
The child can devise their own trail through the snake, taking responsibility for accuracy. If, say, 20 functions are expected, then it is possible to specify that 4 will be +, 4 as -, 4 as * and 4 as /.
The level of challenge can be differentiated, by expecting somewhere in the snake for numbers to go above a specific point.
Whatever happens through the snake, the answer at the “head” is always zero.
This activity is very simple to organise, but can be adapted to any age and ability, by varying the expectations.
More to come soon...
Create mental images before mental maths.
Early learners need security, which comes from engagement with the real world, interpreted as concrete materials, then as visual models, then into mental images which can be manipulated with facility and accuracy.
There is a significant difference between being able to count and understanding maths, but sometimes you can be forgiven for thinking that it’s that simple. Visiting grandchildren recently, I was delighted when one of the two year old twins took my hand and articulated the counting numbers and went past ten. I was amazed we went past three to be honest, but she had a pattern and it had been retained and could be repeated, just like a poem or nursery rhyme. We sang lots of those too, many embedding numbers and she and her sister joined in.
This set me to thinking though, about early number acquisition.
There’s the stage of knowing that number and size exist as a descriptor of the world. Things exist as a group, the cardinality of a number; it has some value, which doesn’t change, ie the conservation of number. This links with 1:1 matching. Things can be put in order, the counting numbers, representing one more, or they can be ordered as fist, second, third. They can be compared, more/greater than, less than (fewer than), heavier/ lighter than, bigger/smaller than, taller or longer/shorter than…… Ordinality…Ordering and organising in this way are often a part of play. Real life situations, encountered every day. Maths is everywhere and can form a significant part of everyday conversations. It is certainly a part of the Early Years Foundation Stage and into KS1.
At some stage the child moves beyond 10. This is a point of issue for me, in that for many children, this perpetuates the counting aspects of maths, whereas there is potential for exploration of issues that move towards an understanding of place value.
Put a handful of counting material on a table and ask children to estimate how many there are. Accept all answers, but ask for clarification of their thinking. Now ask them to find out how many there are. Many will just count them up, one by one. It is an interesting exercise to ask the children afterwards how they could prove their answer. Their only strategy is likely to be recounting. However, if some might organise them into groups, showing some order and organisation, they may be in a position to demonstrate that their organisation allows them to count up in bigger units. So 26 may be described as two groups of ten and six units or singles. This, to me is a precursor stage to written modelling, as the image of the two tens and six units can be recorded as an image.
This imaging allows a child access to many of the next steps of mathematics, but the lack of it can relegate a child to numbers swimming around their heads. Without clarity of visualisation, mental manipulation cannot easily occur.
So, I would argue for holding onto appropriate concrete apparatus for as long as necessary for a child to easily access the concepts and to be able to show a facility with thinking mentally. This can develop into a proof after mental activity.
Working with concrete apparatus, modelling and images can underpin all stages of mathematical thinking, with children being allowed to record their findings in ways that are suitable to them. These can be translated into mathematical speak by the teacher or other adult, to demonstrate the links between the images and pure number. This too is an essential stage. It does not always arise by osmosis, apart from a few possibilities.
Numbers do stuff; we have the four “Rules of number”, addition, subtraction, multiplication and division, each of which has a number of synonyms.
Add; put together, add, add together, total, count, count up, figure up, compute, calculate, reckon, tally, enumerate, find the sum of, amount to, come to, run to, number, make, total, equal, be equal to, be equivalent to, count as…
Subtract; take off, take away, minus, reduce, fewer, (difference) decrease, deduct..
So even within supposedly simple experiences there can be vocabulary induced issues, which cannot be ignored, as you can’t determine that children will always experience a maths problem in the way that they have learned to answer it.
Difference, to me, can be taught as addition or subtraction, whereas quite often it is seen as synonymous with subtraction, hence the brackets.
What is the difference between 6 and 14? This problem can be seen from either end. What needs to be added to 6 to make 14, or what needs to be taken from 14 to make 6? It can be made visual with the aid of “towers” of counting material and compared before calculating. Of course the question could be asked differently. How much bigger is 14 than 6 or how much smaller is 6 than 14.
All of these questions are based on the same premise, that 6 and 14 are not the same….
and that’s the challenge for early stage learners; they have to make sense of the language and the embedded concepts in order to be able to think of them for themselves.
So, my early maths recording books would be blank pages, with printed and laminated guide sheets beneath to support organisation. I’d allow children to draw their thinking and get them to talk the maths, recording as needed to interpret, or just making notes as aides-memoire. In the same way as some exams ask children to “show their working”, I’d encourage this approach at all stages. This approach allows insights into the processes, rather than just the right or wrong answer. That can come later, when a child is secure and just needs some practice to prove or improve accuracy.
Check out the Inquiry Maths website, which has a large number of interesting approaches to Maths teaching.
Long career in education, classroom and leadership; always a learner.