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Operation Maths is a pioneering new maths programme for junior infants to sixth class.

Written by a team of six experienced teachers, Operation Maths is built on a concrete, pictorial, abstract approach, or CPA approach, (based on Jerome Bruner’s conception of the enactive, iconic and symbolic modes of representation) which research has consistently shown to be the most effective instructional approach to enable students to acquire a thorough understanding of the concepts required.

This blog post, and future posts, will explain some of the various features of the Operation Maths programme as well as outlining further ways in which this programme can be used to its full potential to enable your students to truly understand maths, not just do it!

Background & Research

As authors, we researched, and were inspired by, the maths books and schemes used in those countries which are the highest-ranking internationally in relation to attainment in primary maths, for example Singapore, Hong Kong, Japan and Finland.

We also looked at best practice in New Zealand, Australia, Great Britain and the United States, as well as the recommendations of our own home-grown publications including the PDST handbooks, NCCA publications (e.g. Bridging Guidelines, Assessment Guidelines etc.) and programmes such as Aistear and Mata sa Rang.

Finally, this was blended with the requirements of our primary school curriculum, in order to create a scheme that is truly innovative in its approaches and strategies and the most forward-thinking maths programme currently available for the Irish market.

FREE access to all of the Operation Maths digital resources on edcolearning.ie, including ebooks, editable plans, and a whole suite of custom made videos and eManipulatives which greatly enhance the teaching and learning experience for both teachers and pupils.

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Chance is one of the most fascinating areas of primary mathematics, since it is concerned with the outcomes of random processes. Thus, the conceptual foundations for areas of mathematics such as probability and combinatorics, can be found in this strand unit.

The big ideas about Chance:

When considering random events and/or processes, we can use what we know (eg past experience and/or knowledge of the variables involved) to estimate/predict the likely outcome(s).

If we identify all the possible outcomes in advance, we can refine and/or express our prediction using mathematical language.

However, no matter how accurate the mathematical prediction, the actual outcome(s) is not certain (except in the unlikely case where there is only one possible outcome); that is the element of chance!

If we collate the results from repeated identical investigations of a specific random process, the actual outcomes (experimental probability) are more likely to reflect the original mathematical predictions (theoretical probability).

Predicting Outcomes: Terminology

When beginning to discuss and predict the likelihood of various outcomes, the initial focus should be on the language of chance, and the terminology that accompanies it.

It can be very useful for the children to identify the various terminology, to discuss their interpretation of it and to explore the contexts in which the terminology is used in everyday parlance.

And while some of the phrases are more objective (e.g. impossible, never, certain, sure, definite), much of the language can be more ambiguous and is open to personal interpretation (possible, might, there’s a chance, (highly) likely, (highly) unlikely, not sure, uncertain).

FACT: To avoid ambiguity, some organisations have agreed on a consensus that equates this terminology with a fractional expression or percentage; you can view one such consensus here.

It can be helpful to try to organise this language across a continuum for the children to interpret and establish their meanings in relation to the other phrases. Ask the children to identify terminology that is used when describing the likelihood of something occurring. Use questions/statements to elicit from the children the vocabulary for chance that they already have; this can be the language that they would use to answer the questions from the text above or could be from their responses to questions such as the following:

What is the chance that it will rain today?

What is the chance that it will be hot today?

What is the chance that it will be dark tonight?

What chance does my team have of winning the league?

What chance does my county have of winning the All-Ireland Championship?

Ask the children to write this terminology on pieces/slips of paper. Sort the pieces of paper into groups and/or order them along a line (continuum), as shown in the images below, with words that have similar or identical meaning together.

This task is a perfect example of a low threshold, high ceiling task, in that all children can participate and there is no limit to the complexity of terminology that can be incorporated. If mathematical values such as percentages and/or fractions (eg 1 in 2 chance) are suggested, the children should be encouraged to incorporate these, as they see fit.

Indeed, in fifth and sixth class the children should be encouraged to use a continuum which is graded from 0-100% and/or 0-1, and to associate and align the vocabulary with mathematical values (eg impossible/never =0%, might or might not/even chance = 50%, definite/certain = 100% etc).

Predicting Outcomes Mathematically

Irrespective of whether it is tossing a coin, rolling a dice, spinning a spinner, picking from a bag, choosing a card, etc., the children should always be encouraged to identify all the possible outcomes, to predict outcomes that are more or less likely, and to justify their predictions.

From Operation Maths 5

The children can also be encouraged to make more mathematical predictions based on their understanding of the variables involved e.g. if we repeated this investigation 30 times, how many times would you expect each colour would be picked? What about 60 times? 120 times? Express the fraction of the total number number of “picks”, that you would expect for each colour. Can you express any of these as a percentage?

When predicting the outcomes of random processes that involve a combination of variables, it can be very useful to use a type of pictorial structure, such as branching (NB these can also be referred to as tree diagrams), to illustrate the possible outcomes. For example, when predicting the outcomes of a double coin toss, children will often think that each of the three outcomes have an equal chance, when in fact there is double the chance (ie 2 in 4 or 1 in 2 chance) of getting a heads and tails combination, than either both heads or both tails (see diagram below).

From Operation Maths 5From Operation Maths 6

However, it is worth noting that, unless the children come up with a similar structure to predict outcomes of combinations, it is preferable to hold back on showing such a structure until they have conducted an investigation, similar to above, where their predicted outcomes did not align to the actual outcomes.

Conducting the investigations

Once all appropriate predictions have been recorded, we can move on to the most exciting part, the investigating! When conducting chance investigations, it is important that the children recognise that that they need to be conducted fairly and recorded clearly, similar to scientific investigations.

Encourage the children to consider what factors need to be kept the same each time, and how practices could affect the reliability of the results eg:

When picking items (eg cubes from a bag, cards from a deck) does the chosen item need to be returned each time? Why/why not?

How many times does an investigation needs to be repeated in order to get a reliable result?

To generate sufficient data, while not spending too much time on each investigation, ten can be a suitable number of turns per child. It can also be a good idea to organise the children into groups of three with rotating roles eg the first child has their turn, the second child records the outcome of each turn and the third child keeps count of how many turns the first child has had, and roles are rotated after ten turns.

Recording and reflecting on results

As mentioned previously, the children should be encouraged to consider how best to record results. Tally charts and frequency tables can be very useful and link in well with the strand unit of Representing and Interpreting Data. Results of investigations can be displayed in various types of graphs and charts. Children in fifth and sixth classes could also be asked to calculate the average value for each outcome, when all the results of a class group are considered; for example, in the double coin toss, what was the average number of heads, tails and heads-tails combination per group.

Once the results have been collated, it is very important that the children be given time to reflect on the results and to compare them to their predictions. While we would expect an equal number of heads and tails in a single coin toss (ie theoretical probability), the actual results may not resemble these predictions (experimental probability). Such is the element of chance! And this can be a difficult concept for the children to accept, particularly the notion that, even though the mathematics behind their predictions was accurate, the actual outcomes are different.

To explore this further, using a spreadsheet, such as Google Sheets or Microsoft Excel, to collate the results of the entire class can be a great way demonstrate, that when we combine all the investigations, experimental probability (ie the results) is more likely to mirror theoretical probability (the predictions). This can often help reassure the children that the “maths” behind this does indeed work!

Further Reading & Resources

Playing dice, card, spinner games, or indeed any type of chance-based games, can be a great way to get students thinking about probability, while also providing practice with mental computation, estimation, subitising and experience of problem-solving via strategic thinking.

Don’t forget to check out the games bank in your Operation Maths TRB and/or the last page of the Number Facts books for examples and ideas.

iTools has a great set of interactive tools for probability that cover coin and dice throws, pulls from a bag, among other random processes. As well as being very customisable, they compare the theoretical and experimental probability, using various visual structures including tables and branching (tree diagrams); the latter is used particularly well to illustrate possible outcomes in compound events (e.g. double coin toss or double dice throw) as well as combinations and arrangements.

For a fifth and sixth class who are exploring combinations, Mashup Math has two excellent videos (view both below) which demonstrate how tree diagrams and area models can be used to identify all possible combinations; both video use contexts to which the children could readily relate.

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Welcome to the May installment in this series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

For more information on dedicated primary events and resources check out the Tech Week site.

Outdoor Classroom Day is May 23 2019. This global event encourages us to use the outdoors to teach, explore and learn. There are lots of resources with suggestions for all subject areas, including maths, https://outdoorclassroomday.com/. For more ideas for outdoor maths you could also check out:

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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Welcome to the April installment in this series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter

Pssst! There are still some Edco Primary Publications launches for 2019 taking place during April; specifically in Sligo, Dublin North, Kildare, Castlebar, Galway, Waterford and Athlone. As well as launching the new SESE programme Explore with Me, the new English Core Skills Let’s Talk Literacy and Bua na Cainte 4, they will also be showcasing Operation Maths, Number Facts and other Edco publications. Click on the link above for more information and to register.

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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Welcome to the March installment in this series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter

Pssst! The Edco Primary Publications launches for 2019 will be taking place around the country during March and April. As well as launching the new SESE programme Explore with Me, the new English Core Skills Let’s Talk Literacy and Bua na Cainte 4, they will also be showcasing Operation Maths, Number Facts and other Edco publications. Click on the link above for more information and to register.

Second Class: Operations (addition, without and with renaming, and subtraction without and with renaming, the latter of which is new content while the other material will be revision of first class work); Length (including the formal introduction of the centimeter as a standard unit of measure).

Other suggestions for March:

Engineer’s Week runs from Saturday 2nd to Friday 8th March. There are obvious connections between Maths and Engineering, a fact which is being celebrated by the STEM (Science, Technology, Engineering and Maths) movement globally. Click on the link above to download a primary school challenge pack which contains lots of ideas to help you organise fun challenges that create a positive awareness and spark enthusiasm about the engineering profession in young people.

Of the STEM areas, coding is one of the most exciting, not least of all to kids! And Operation Maths is the only Irish maths programme that has integrated coding activities via the Scratch Lessons for Operation Maths 3-6. Check out the scratch lessons that are included in the Operation Maths digital resources via your TRB or edcolearning.ie

For infants, the Aistear Themes are an ideal way to explore STEM using a thematic focus; consult the Junior and Senior Infants TRB for the monthly Aistear suggestions.

For some more primary-focused STEM activities, check out the links below:

Pancake (Shrove) Tuesday is Tuesday 5th March. Recipes naturally provide great opportunities for real world maths, for example identifying the measures and amounts required, adding the correct measures to the mix, adapting the recipes to suit more or less people, etc. For more maths-related activities check out these pancake problems.

World Book Day is on Thursday 7th March, and while the primary resources accessible on the site are mainly literacy linked, there are numeracy ideas also, including “Not Another Maths Book” activity sheet. Other numeracy and literacy linked suggestions for this global celebration include:

Carry out a survey to find out the favourite books / authors of the children in your class. Or choose a page from a book. Work out the average number of words per sentence. Both of these are included among many other suggestions from Teaching Ideas for World Book Day.

With both Ireland’s ultimate and penultimate game in the 2019 Six Nations still to be played, there is still time to delve into some of the mathematical possibilities:

Calculate the number of games to be played; what if the competition had less or more teams, how many games would need to be played then?

Use the language of chance to discuss the possible outcomes for each nation in the competition and recognise that while it is impossible to predict the actual outcomes, we can use of knowledge of the teams performances to make informed predictions.

Make score predictions for each match and plot how these scores would be recorded on the Six Nations Table

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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3-D shapes or 3-D objects?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe both 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is titled 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the 1999 Primary Maths curriculum.

So what first? 2-D or 3-D?

2-D and 3-D objects are very inter-related, to the point that there is often much debate about which of the topic should be taught first; 2-D shapes, 3-D objects or teach them both concurrently.

Since 2-D shapes are lacking the third dimension of
depth or height that their 3-D relations possess, this makes them quite
abstract as only flat, drawn/printed shapes are truly 2-D. Whereas, 3-D objects
can be picked up, manipulated, used for constructions etc., making them much
more suited to the concrete learning experiences that are essential in the
early years. They are the objects that we find in the real-world. Thus, since
2-D shapes are only flat representations of the faces of 3-D objects, it could
be argued that it would be more logical, and more in line with the concrete-pictorial-abstract (CPA)
approach, to teach about 3-D objects before
2-D shapes.

On the other-hand, it could be argued that 2-D
shapes should be taught first as it is likely than young children would be more
familiar with them. For example, the vocabulary of 2-D shapes features more
regularly in common speak than the vocabulary of 3-D objects. Many children
will likely have encountered many 2-D shapes from picture books and patterns
around their homes, etc. And so, it remains inconclusive as to which order of
progression is most beneficial!

In the Operation Maths books, the children meet the
specific topic of 2-D shapes prior to that of 3-D objects each school year.
However, it is envisaged that by the time the children in the junior classes
are formally engaging with 2-D shapes, they have already encountered and informally
explored both 2-D shapes and 3-D objects via the monthly themes (laid out in
the long-term and short-term plans of each TRB) and in the suggested Aistear
play activities (detailed also in the TRB) of which, the Aistear theme of
construction is particularly relevant.

Infant classes

Whether considering 2-D shapes or 3-D objects, the suggested progression within each topic is very similar:

Undirected play

Sorting and ordering activities

Building and making (including making patterns)

Identifying

Undirected play may include sand and water play, use of formal construction toys, constructing using “junk” or found materials; any activities that allow the children to handle and manipulate shapes and objects. In the Operation Maths TRBs for junior and senior infants there are ample suggestions for suitable activities, under the headings of various themes. “Undirected play” does not imply that the teacher is superfluous to the process; rather while the children are the instigators, the teacher can play a vital role, observing the way in which the children interact with the materials, and asking the children to explain what they did, how they did it and why they did it that way. This can be a great way to assess the prior knowledge and language that the children may already have.

Sorting and ordering activities include the Early Mathematical Activities (EMA) used early on in the infant classes; thus it is likely that shapes and/or objects have already been used as part of these activities, for example sorting and matching according to colour, size etc; ordering according to length/height etc.

At this point, the children should also be prompted to sort the shapes and objects according to their respective properties as relevant and appropriate:

Sort 2-D shapes according to the number of corners and the number and type of sides (straight, curved or both; sides that are different or the same).

Sort 3-D objects according to those that roll/do not roll, slide/do not slide, build/do not build, are hollow/solid; as a development, according to the number of corners and the number and type of faces and edges (please see end of post for more information on faces and edges).

The teacher can also isolate shapes/objects to create sets and then ask the children to identify the rule of the set: “What’s my rule?” (see image above). The children can also be encouraged to play the “What’s my rule?” game in groups.

Isolate a particular shape/object in the room and ask the children to locate others that are the same/similar and make a set of like objects/shapes.

The children may also be naming the shapes as part of these explorations; however this is not necessary as it is more important that they appreciate the similarities and difference between shapes, rather than identifying them.

Building and making with shapes/objects may have already been explored informally as part of the undirected play phase. The purpose now is to develop this into more formal teacher-directed tasks and activities:

Build the tallest building/castle that you can. What objects did you use/not use and why?

Dip a face of a 3-D object in paint and use it to print. Make a pattern using the prints. What do you notice?

Try printing with different faces of the same 3-D object. Are the resulting prints the same or different?

Push 3-D objects into sand/plasticine to make imprints. Or (if able) trace around the 3-D objects on paper to make designs.

Use the shapes to cover surface of your book/mini-white board; which shapes did you use/not use and why?

Combine two shapes/objects to make a new shape/object.

As part of the building and making activities the children may begin to realise how certain shapes/objects can be combined to become other shapes/objects. Similarly, through the reverse of these activities, and other shape cutting activities, the children should begin to realise that shapes can also be separated (partitioned) to reveal new shapes. This can include deconstructing 3-D objects to reveal their net. These activities can be revisited once the children can also name the shapes/objects, so as to arrive at certain understandings and become more accurate with mathematical language eg that two squares can make a rectangle; that, when using tangrams, two of the same size triangles can be rearranged to make a square, a larger triangle etc.

Identifying the specific shapes/objects evolves from the previous activities as the children begin to realise that it is the specific properties and attributes of a shape/object that defines it, eg all shapes with three corners (and three sides) are triangles, irrespective of their size or colour and irrespective of the measure of sides and corners (later to be referred to as angles). Activities which will serve to reinforce this include “Guess the shape/object” using descriptions (see below), guessing unseen shapes/objects from touch (eg in a feely bag), locating a specific shape from a collection using touch alone. Through the experiences of printing and imprinting with the 3-D objects, it is also hoped that the children realise that the flat faces of 3-D objects are in fact 2-D shapes.

First and Second classes

The children in these classes will continue to sort, describe, compare and name shapes as done in infants, but to now also include new shapes and objects i.e. semi-circle (1st), oval and cone (2nd). They will continue to construct and make shapes, extending this to creating and drawing the shapes themselves.

They will further explore the combining and partitioning of 2-D shapes, and this understanding will extend to include the fractions of halves (1st) and quarters (2nd). The properties of 2-D shapes will be further explored in second class via the stand units of symmetry, angles and area (i.e. tessellating 2-D shapes)

Q: How many faces on a cylinder? Three or two?Traditionally, in Ireland, and in Irish textbooks, a cylinder was recorded as having three faces. However, this is not mathematically correct, as strictly speaking a face is flat, and a 2D shape (figure), so therefore a cylinder has in fact only two faces, (both circles), and one curved surface. And while it may be argued that a cylinder has a third face i.e. the rectangular shape you see when you disassemble the net of the 3-D object, in this disassembled state it is no longer a cylinder, since it can no longer roll, a specific property of all cylinders. Another way to think about the faces of 3-D objects is to consider the number and shape of the resulting outlines of tracing around, or printing, each surface of the 3-D object. It is only possible to trace around the opposite ends/bases of the cylinder, since only these are flat, thus it has only two faces, both of which are circular in shape. Similarly, it is only possible to trace around one surface on a cone, which therefore means it has only one face (a circle) and one curved surface. And how many edges on a cylinder? Officially none, as an edge is where two flat faces meet and the faces on a cylinder are on opposite sides and do not touch/meet. However, that leaves the problem of how to describe the place where each face meets the curved surface. So in Operation Maths, as occurs typically in other primary texts in other countries, there is a distinction made between straight edges (which are in fact true edges) and curved edges (which strictly speaking are not edges).

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Welcome to the February installment in this series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter

Second Class: Weight, Time (to quarter hours on an analogue clock and to half hours on a digital clock) and Estimation (how to estimate to total of addition calculations within 99). The development of keen estimation skills is very important throughout Operation Maths and work here will lay future foundations.

Calculate the number of games to be played; what if the competition had less or more teams, how many games would need to be played then?

Use the language of chance to discuss the possible outcomes for each nation in the competition and recognise that while it is impossible to predict the actual outcomes, we can use of knowledge of the teams performances to make informed predictions.

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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And welcome to the fifth installment in this series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter

Other suggestions for January:

Storytelling Week runs from Sunday 27th January to Sunday 3rd February. While this is primarily a UK based event, it does serve as a timely reminder of the rich role that mathematical stories can play in the early years. For teachers of infants to second class, be sure to check out the Literacy suggestions within the Integration section of each short term plan in the TRB.

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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Of all the strand units in maths, this topic is one that is very close to the hearts of almost all young children:

“She’s got more than me! That’s not fair!”

“I want to be first!”

“I want to be the biggest!”

This strand unit evolves from the separate strand units of comparing and ordering that, along with the other two strand units of classifying and matching, make up the strand of early mathematical activities. The content objectives for this strand unit are quite similar across the four junior classes, with the main difference being the specific number limits for each class level:

Number > Comparing and Ordering >The child shall be enabled to:

compare equivalent and non-equivalent sets (to include the symbols <, >, = in second class)

order sets of objects by number (infants to first only)

use the language of ordinal number

Comparing

As mentioned above, even from when they are very young, most children are quite adept at comparing what he/she has with that of another.

As part of the strand early mathematical activities (i.e. pre-number) the children will already have had experience comparing sets by quantity (but without counting) i.e. identifying which of two sets has an obvious amount more (or less) than another. They will also have been identifying two sets/objects as being the same or different.

In Junior Infants, once children are comfortable establishing the cardinality of sets up to five, the next step is comparing and ordering sets of objects up to five. Since the amount of these sets may often only differ by just one or two, then it is not very obvious, from a visual point of view, which one has more or less. Comparing two similar sized sets requires that the child:

Can identify (and, later, write) the correct numeral for that set

Understands one-to-one correspondence, and using this can match the items in the two sets, so as to establish which one has more or less

Understands the conservation of number i.e. that a short line of five objects situated close together still has more than a longer line of four objects further apart.

Does not assume that the quantity of a set with objects bigger (or smaller) in size must be greater (or less) than the other set.

How many more?

Once a child is able to identify the greater set, the next step is to be able to state the difference between the sets i.e. how many more plates than cups? This can be a very difficult concept, with which children can struggle for many years.

As with the entire Operation Maths programme, a CPA approach is recommended when teaching this concept and, in particular, to use that which is most familiar to the children:

Use items that typically go together eg knives and forks, cups and saucers/plates, children and chairs/coats/school bags. Take a number of each and ask the children to suggest how we could ascertain the number of each. If not siggested by the children, the teacher should demonstrate how to set out the items in groups toegther eg the first knife with the first fork, the second knife with the second fork etc. If the quantities of each are not equal/the same, ask the children to explain how many more of the lesser quantity is required AND to explain how many extra items there are in the larger amount.

In a mixed classroom, use girls and boys. Call up a random group of children, ask the boys to line up at the top of the room, and the girls to line up in separate line beside them, so that, where possible, each child is adjacent to one other child in the other line (if you are lucky enough to have square tiles on your floor, ensure that there is a child standing in each square space). Ask the children to identify the children who have a match/partner on the other line and the number of children who do not have a match/partner on the other line. This activity could also be repeated using dolls and teddies, toy farm or zoo animals, attribute bears etc.

Use concrete manipulatives and pictures. Start with only two sets initially. Impress up on the children that the easiest way to see the comparison is to “line up” the objects, was done with the children previously. Use a grid of squares* to help with this. Once again, ask the children to identify where there is a “partner” fruit on the other line and the number of fruit that do not have a “partner” on the other line. These are the extras. How many more (extra) bananas than apples? How many more (extra) bananas than strawberries? *The 5×5 grid on the Operation Maths Sorting eManipulative is very useful here. The Operation Maths 100 Square eManipulative can also be used; select to show counters only and line up two (or more) rows or columns of different colours.

Ultimately, it is hoped that the children realise that it is not necessary to establish the exact amount of each set to be able to establish the difference between each set. In the example above, there are two more bananas than strawberries, and it is not necessary to identify the number of each fruit to establish this. This encourages the children to develop efficiency and flexibility in their approaches.

As the children move into first and second class, they should still be encouraged to “line up” the sets. If comparing the number of items in two static sets that cannot be lined up, eg an image in their books, the children can represent the number of items in each set using cubes and these cubes can then be lined up to make it easier to identify the difference between each set. This would link very well with their experiences of comparing quantities in pictograms and block graphs from the strand of Data.

It is important that teachers are aware that establishing the extra number in the larger/greater set and establishing how many less/fewer in the smaller/lesser set requires the children comparing the amounts in two different ways. In the example above, to identify how many more bananas there are than strawberries, requires identifying the number of bananas for which there are no corresponding strawberries. However, to identify how many fewer strawberries there are than bananas, requires identifying the number of empty spaces in the strawberries that there are, opposite the extra bananas. While the answer is the same both time, the route to the answer is different, and the latter approach requires the children to count empty spaces, which is more challenging due to its abstractness.

In second class, the children will begin to use the inequalities symbols (<, >). Many children will struggle with selecting the correct symbol to use, even if they can identify the larger or smaller quantity. Thus flashcards or reference cards such as the ones at this link can be very useful to connect both the language and the symbol. Interactive quizzes like this one from That Quiz or this one from ixl.ie can provide opportunities for extra practice. However, as emphasised previously, it may still be necessary to use a visual representation of both numbers being compared, for example using stacks of cubes, base ten blocks, straws or base ten money (10c and 1c coins). In this way, the children are now beginning to use their place value understanding also to compare quantities. As well as using the actual concrete materials, the Sorting eManipulative can be used to demonstrate how to do this using images of base ten materials; see Ready to go activities 2.3 and 2.4 as examples (screenshots below).

Hint: Developing the children’s ability to compare, will also be of benefit when they encounter the concept of subtraction as difference (as opposed to subtraction as deduction/take-away) and of further benefit when they are introduced to comparison bar models in third class up

Ordering

As part of a early mathematical activities, the children will already have experienced ordering objects by length, size etc. Now, they are extending this understanding to order by quantity.

In Junior Infants, once the children are able to count individual sets of up to five objects, this enables them to start ordering the sets of objects.

The children are beginning to understand how higher numbers correlate with greater numbers of objects and vice versa.

When ordering sets we must also consider the number word sequence i.e. number five comes after the number four so five must be a greater amount than four.

Ordinal numbers

The nature of the English words for the ordinal numbers (first, second, third, fourth etc) and the nature of their abbreviated forms (1st, 2nd, 3rd, 4th etc) can pose significant difficulties for children as, at first glance, there appears to be little correspondence between the forms, and the abbreviations may not appear to follow any rule or pattern. Another difficulty lies in the apparent contradiction between ordinal numbers and cardinal numbers; it is typically better to have 10 rather than 1 of anything, but it is typically better to be 1st rather than 10th in any competitive activity.

Initially the focus should be on the spoken words only and the activities used should reflect this eg lining up children at the classroom door, asking the rest of the class to identify who is first, who is second, third, last etc.

When ready, flashcards of the ordinal words should be introduced and these can be incorporated into the familiar activities eg the flashcard with “first” can be given to a child who must give it to the child in that position in the line.

It is better to avoid using the abbreviations until first class and it is also better to start with the words, fourth, sixth seventh and tenth. Write the word fourth on the board and establish that the children can read and understand the word. Explain that for speed we want to find a quicker way to write/indicate this position and ask them to suggest what might be written to replace the underlined part of the word (ie 4th). Repeat this with the ordinal words sixth, seventh and tenth. Ask the children to suggest how fifth, eighth and ninth might be abbreviated and then finally ask for suggestions for the words first, second and third; ultimately, tell them the correct answers if they do not arrive at them themselves. In this way, the children are being prompted to discover the system of abbreviations that we use, as opposed to being just told.

Hint: For first and second classes, there is a list of online interactive games here which will help as extra practice. There are also lots of useful videos on YouTube etc; just search for “ordinal numbers”.

The Operation Maths Digital Resources have specific resources designed to support this strand unit, and have suggestions of online games in the Weblinks section. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.

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At Operation Maths we are constantly looking for ways to improve the usability of our programme, and to make it even more teacher-friendly. The most recent additions included long term plans for various combinations of multi-classes as well as excel versions of our Assessment Records. Following on from feedback from teachers, we have now expanded the original version of the latter, released in October last, to include all the assessments for the entire school year, as well as incorporating extra features designed to make them even more teacher-friendly.

While there are already word versions of these assessment records available to download from Edco Learning, as well as the hard copy photocopiables in the Teacher’s Resource Books (TRBs), these excel versions provide teachers with a more efficient and flexible way to both record and analyse the results from the Assessment Booklets:

Quickly get a total attainment score for each child (Assessment of Learning)

Use these attainment scores to compare the attainment of various individuals and/or groups of children and identify children in need of further support (Assessment for Learning).

Quickly get a score for each learning outcome, use these scores to identify the strengths and weaknesses of the class as a whole, while also being able to identify which learning outcome(s) require further consolidation (Assessment for Learning).

And this is all achievable in a very teacher-friendly way:

Each of the five assessments ( eg End of October, End of December etc ) has a dedicated page; click the tab at the bottom to move between them.

Teachers need only enter the children’s names once only on the first page; the inbuilt formulas then copy these names to the other pages in the document.

Under each child’s name, the teacher can enter a score for each question (or page in junior infants); see more below for a suggested scoring system.

The score for each individual question (or page) will be automatically totaled (horizontally across bottom) to give an attainment score for each child.

When all the scores have been entered for each child, these will also be totaled along the right-hand side vertically to give a total for each learning outcome.

After the five assessment tabs, there is a tab entitled “All”. Here all the scores from each assessment will be automatically replicated, once entered on the original assessment tab. This allows the teacher to easily view all the data in one screen. The scores for each child will also be totaled here to give you an overall score.

Other useful information provided includes the specific strand and strand unit (S.SU) to which the learning outcome relates. These are abbreviated and a full explanation of the abbreviation is given on the last tab.

HINT: If you already used the version of this document released last October, it is now recommended that you download and use these documents for the remaining assessments. They also include a page for the End of October Assessment, allowing teachers to copy and paste in the October Assessment scores, so as to have all the information in the one document.

Suggested Scoring System

While teachers can devise and use any system which they prefer, one option would be to try the following:

4 = Question answered fully and correctly

3 = Question answered fully but without full accuracy ie almost all correct

2 = Has a majority of correct responses but a number of errors also

1 = Some correct responses but a majority of errors

0 = Not attempted or incorrect responses

Obviously, teachers will have to apply any scoring system in a flexible way; for example if there is a question that requires just one response and is therefore is either correct or incorrect, then only 0 or 4 will be awarded.

Once the appropriate score has been entered for each question then the teacher will have:

a total attainment score for each child; the higher the score the more learning outcomes achieved.

the means to sort and/or compare the attainment of various individuals and/or groups of children using these total attainment scores and identify those children with the lowest scores as needing further support.

the means to to identify the strengths and weaknesses of the class as a whole by sorting/comparing the total scores for the learning outcomes, while also being able to identify which learning outcome(s) would benefit from further teaching.

Downloading and using

The excel documents for each class level are available to download by clicking on the links below:

We welcome all feedback!
And it doesn’t have to be specific to these assessment records. Remember, that if you have any suggestions or any questions on Operation Maths, Number Facts or anything related to primary maths, please PM or contact Edco Primary Maths via Facebook and/or Twitter

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Spatial awareness…being able to describe the position of something/someone in relation to another using words and/or gestures, and being able to represent spaces and locations using models and/or drawings, may, at first glance, appear to have more in common with communication and geography, than with maths. However, the concepts of spatial awareness lay the foundations for all geometric thinking, be it at upper primary, secondary or an even higher level.

Essentially the children need to develop an understanding that:

The spatial relationships between objects and places can be described and represented.

These relationships may be viewed, described and represented differently depending on the perspective of the viewer (in particular, consider left and right).

Developing the ability to mentally visualise the representations will enhance a person’s ability to picture how a shape will look when rotated when turned, flipped etc.

A synopsis of the curriculum objectives for infants to second class, state that the children should be enabled to:

explore, discuss, develop and use the vocabulary of spatial relations (describing both position and direction/movement)

explore closed shapes and open shapes and make body shapes

give and follow simple directions (first and second class), including turning directions using half and quarter turns (second class only)

explore and solve practical problems (first and second class)

In the case of the practical problems, this could include completing a jigsaw or a tangram puzzle, using mazes, grids, board games and or exploring basic coding eg via coding programs and apps, such as Lightbot, and more hand-on devices such as BeeBots.

Moving through space

Since spatial awareness requires an understanding of using space and moving through space, the majority of the activities should be active ones, where the children are moving around. This is where the suggested activities in the Operation Maths Teachers Resource Book (TRB) become extremely useful, such as the examples below.

Much of the language development in this strand unit can be reinforced via activities in PE (Orienteering) and Geography (mapping).

Digital Resources

While activities incorporating physical movement are preferable, the Operation Maths digital resources on edcolearning, provide a worthwhile alternative and add variety. The Ready to Go activities below, as the phrase says are “exactly what they say on the tin”; the teacher need only click on the relevant icon in the digital version of the pupil’s book to open the activity, and the accompanying suggested questions are quickly viewable along the side menu. A full description of the activity, including the questions, is also given in the TRB.

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