Author Archives: Operation Maths

A CPA approach to Maths

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Category : About Operation Maths

As explained in previous posts, Operation Maths is built on a concrete, pictorial, abstract approach, or CPA approach. Developed by American psychologist, Jerome Bruner, it is based on his conception of the enactive, iconic and symbolic modes of representation. Research has consistently shown this methodology to be the most effective instructional approach to enable students to acquire a thorough understanding of the concepts required. This CPA approach is also the mainstay of maths teaching in Singapore.

What exactly is CPA?

Concrete Pictorial Abstract (CPA) is a three step instructional approach that has been found to be highly effective in teaching math concepts.

  • Concrete stage: Also known as the “doing” stage, this involves physically manipulating objects to demonstrate and explore a concept.
  • Pictorial stage: also referred to as the representational stage in some literature, it can be explained as  the “seeing” stage and involves using images to represent the objects previously used in the concrete stage.
  • Abstract stage: also known as the “symbolic” stage and involves using only numbers and symbols to represent and solve a computation.

What does CPA look like?

Below are some examples of a how a CPA approach might look, at each of the main  class levels:

Concrete Pictorial Abstract
Infs Use logic bears, toys, etc to show a set of five and explore the various ways to partition and then re-combine Use counters and ten frames, cubes , blocks, cuisinere rods and/or draw images to represent the concrete Use digits and/or symbols to represent the relationships established during  the previous two stages eg 2 + 3 = 5
1st/2nd Adding and subtracting, without or with renaming using base ten materials eg straws, cubes, base-ten blocks Use or draw images to represent the concrete manipulative Use a written algorithms for addition and subtraction
3rd/4th Explore multiplication using rows of base ten blocks (area model of multiplication) Draw images to represent the concrete manipulatives Use a written algorithm for multiplication
5th/6th Explore operations with fractions using concrete manipulatives eg paper plates in halves, quarters, eighths Draw images to represent the concrete manipulatives Use a written algorithm and/or branching

 

What does CPA look like in Operation Maths?

The best way to fully appreciate the CPA approach in Operation Maths is to look at some examples.

Addition without renaming (Operation Maths 1)

The children should use real base ten blocks to model the calculations, before progressing to using the pictorial representations and then, finally, to the column method of the written algorithm

 

Multiplication involving two-digit numbers (Operation Maths 4)

Using base ten blocks to demonstrate multiplication as an area array
Moving on from the actual blocks; drawing a pictorial representation
Moving on from the area models, using grids
Using the partial products method
Ultimately arriving at the traditional algorithm; the abstract stage

 

Adding fractions and mixed numbers (Operation Maths 5)

Suggestions for concrete activities in the Teacher’s Resource Book (TRB)
Examples shown of how to use the concrete materials, as well as showing how branching and number lines could be used
Fraction pie pieces in the Discovery Book; there are also blank number lines as an extra pictorial resources given on the inside cover of the Pupil’s Book

Other examples of a CPA approach in Operation Maths

These are only a few small selection of examples of the CPA approach across the Operation Maths programme. Other examples are:

  • The inclusion of free five, ten and twenty frames with the infants to 2nd class books which enable teachers to include frames as one of the concrete activities.
  • The inclusion of free place value manipulatives with the 3rd to 6th class books which enable teachers to include explore and use these resources to demonstrate place value, addition, subtraction, multiplication and division.
  • The free mini-white boards (MWBs) facilitate the drawing of quick jottings to represent concepts and calculations.
  • The TRBs suggest ways in which the teacher can organise concrete activities and use real objects to explore concepts, including suggestions for stations and Aistear themes in the junior end TRBs.
  • The inclusion of base-ten money as photocopiables in the senior end TRBs i.e. the images of 1c, 10c and €1 coins, €10 and  €100 notes. These can be used to add variety to the resource examples and also provide the means to explore decimal numbers in a concrete way.
  • Within the RUCSAC approach to problem-solving, the stage “create to show what you know” specifically prompts the children to use concrete materials and/or pictorial representations to represent the problem.
  • The use of visual strategies for problem-solving,  such as bar models, number lines (for whole numbers, decimal numbers and fractions), number bonds and branching, also provide a pictorial way to bridge the gap between the concrete and the abstract.
  • Many of the digital eManipulatives, accessible on edcolearning.ie, are themselves pictorial representations of real objects; the sorting and shop eManipulative, the fraction eManipulative, the bar-modelling eManipulative and  the counting stick eManipulative can be all used to demonstrate concepts in a graphic way.
  • The Maths Around us videos also use real life objects to show ways to represent mathematical concepts

Some final thoughts…

My own experience of primary maths was typified by the abstract stage; in maths texts of the time, the exercises and even the explanatory sections were almost entirely digits and symbols based, with little or no visual imagery. In recent times, teachers are more aware of the importance of incorporating concrete activities into maths instruction and do so regularly. However, I do think that this is more evident in the junior classes and that teachers of the senior classes sometimes struggle to find ways to demonstrate concretely the more complex concepts required by the curriculum in those classes.

I also believe that the pictorial stage is often neglected and that instructional activities often jump from the concrete stage straight to the abstract stage. If we think of the three stages as stepping stones on a child’s journey to mathematical understanding, many of the stronger, more mathematically-able children are able to make the leap from the concrete  to the abstract. However, for the less able, this can be too big a leap and they don’t successfully manage the jump. For these children especially, it is vital that we ensure the pictorial stage becomes a regular intermediary part  of the instructional sequence.

Thankfully, teachers no longer have to struggle to come up with ways to represent complex concepts or search for ideas for concrete  and pictorial experiences for their classes; instead, Operation Maths is ticking all those boxes and then some!

Operation Maths: A Unique Approach to Problem-Solving

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Category : About Operation Maths

In this post, we will look specifically at the Operation Maths approach to problem-solving in the senior end books (3rd to 6th classes). In a subsequent post, we will look at how this approach develops in the junior end books (infants to second classes).

Presenting children with an abundance of mathematical problems does not automatically transform them into competent and confident problem-solvers. Rather, the children must be explicitly taught a range of problem-solving
strategies and they must be facilitated in applying and practising the strategies repeatedly in a range of different contexts.
Operation Maths has an integral multilayered approach to problem-solving throughout the 3rd to 6th class books:

  • A variety of key problem-solving strategies is introduced, explored and applied to various real-life contexts in a developmental and spiral way through the classes (i.e. bar model drawing, empty number lines, T-charts , branching etc)
  • Regular Work It Out! sections throughout the chapters in the pupils books provide the children with opportunities to apply and hone their problem-solving skills.
  • Let’s Investigate! sections at the end of the Pupils’ Books where the focus is on open-ended problems
  • Thematic revision spreads with a strong problem-solving focus.
  • Extra problem-solving in Early Finisher photocopiables.

All of this happens as part of a larger problem-solving approach based on the acronym RUCSAC. This approach, which can be used as a whole school problem-solving approach, is also reinforced and explained for both children and parents on a convenient French flap/bookmark on the Discovery Book (see images from flaps below), which encourages the children to use RUCSAC as an aid when problem-solving.

Problem-solving skills

The ability to reason mathematically is fundamental to being able to solve mathematical problems. However, reasoning mathematically requires not just one, but a number of mathematical skills e.g. being able to
• Work through a problem in a systematic way
• Predict an answer
• Identify the relevant information and understand what type of answer is being sought
• Visualise the problem mentally or being able to represent the components of the problem in either a pictorial or abstract (using only numbers and symbols) way.
• Plan or decide what approach to take
• Work to get an answer
• Check that the answer is suitable and accurate.

What is fundamentally different about the Operation Maths approach to problem solving is that the children are being taught specific strategies to develop the aforementioned skills, in a spiral and progressive way, in order to equip the children with the necessary skills for them to become capable and confident problem-solvers.

Central to the Operation Math approach to problem solving is RUCSAC. This clear, sequential approach enables the children to work through problems in a systematic way, while simultaneously utilising the mathematical skills that are being developed with and throughout the chapters.

 

RUCSAC and the Specific Strategies taught in Operation Maths

RUCSAC is an acronym, where each letter represents one of the six distinct phases of this problem-solving approach (see below). However, this more than just a clever mnemonic, as each of these phases is supported by the development of specific strategies throughout the programme, which support this approach.  These specific strategies are as follows:

Read – Estimation strategies:

  • Reasonable answer: Would you predict a bigger or smaller answer? How many digits would you expect in the answer
  • Front-end estimation: Look at the digits at the front of the numbers
  • Rounding: Round each number to the place of the highest value digit e.g. tens, hundreds, thousands.
  • Rounding to fives: (only in OM6): Usually we round to the nearest tenth, unit, ten, etc. But if the number(s) involved are approximately in the middle, it is more efficient to round them to the nearest five tenths, 5, 50 etc. to get a more accurate estimate. (OM6, Pupils Book p 30)

Underline – Colour coding operational vocabulary:

  • Identifying specific phrases, colour coding them, and recording them on in the Discovery Book. This forces the child to engage with the language of problems and to decode them. However, this only suits word problems which contain obvious operational vocabulary or that which can be easily inferred.

Create – Creating visual representation to show the information in the problem, as part of a CPA approach:

  • Using concrete materials (e.g. counters, cubes, children etc.)
  • Using bar model drawings
  • Using empty number lines
  • Using T-charts (OM4 to OM6)
  • Making/completing a table, grid, list etc.
  • Creating number sentences (and/or equations with variables in OM6)

Select – Selecting a suitable and efficient approach:

  • Using a mental method, e.g. petitioning, sequencing, compensating etc.
  • Using a written method e.g. a formal algorithm, jottings, branching
  • Using guess and test

Answer – Answering the question:

  • The teaching panels demonstrate how to layout and position work clearly and sequentially
  • Children are encouraged to “show your thinking”

Check – Checking answer(s):

  • Comparing the answer to the estimate, e.g. does it look reasonable?
  • Using the inverse to check.

Furthermore, as part of this approach, specific visual strategies are introduced and repeatedly used where appropriate:

  • Empty Number lines
  • Bar Models
  • T-charts

 

Empty Number Line (ENL)

Simply, a horizontal line, initially with no numbers or markings that helps develop a child’s number sense, their ability to visualise numbers and to compute mentally.
Also known as a blank or open number line, empty number lines can be used to show elapsed time, operations, skip counting, fractions, decimals, measures, money (making change) and much more (see image below).

While, strictly speaking the number line should initially start empty (i.e. no numbers or markings), in Operation Maths, some of the required numbers and/or markings have been provided, to act as scaffolding for the child. Ultimately, it in envisaged, that as the child grows more confident of this structure, he/she should be able to construct an empty number line from scratch in order to help solve other problems. I is also hope that through using this structure the child would be able to develop this ability to visualise numbers in such a way and, in doing so, enhance their ability to compute mentally.


Bar Models

These are simply drawing(s) that resemble bars, (like that seen in bar graphs), that are used to illustrate number relationships. There are two main types, part-whole bar models and comparison bar models.


Part-whole model: which can represent a whole amount that is subdivided into smaller parts. In Operation Maths these are used to represent:

  • Addition/subtraction: where a whole amount has been subdivided into two or three amounts/parts and either the value of one of the parts or the whole/total is required
  • Multiplication/division: where a whole amount has been subdivided into equal amounts/parts and either the value of one/some of the parts or the whole amount is required
  • Fractions, ratios, decimals and percentages: Where a whole amount has been subdivided into equal amounts/parts and either the value of one/some of the parts or the whole amount is unknown.

Comparison models:  which are used when comparing two or more quantities. In Operation Maths these can be used to represent:

  • Addition/subtraction and Multiplication/division: where two amounts are being compared and the value of one of the amounts or the difference between the amounts or the total value of the amounts is being sought.
  • Fractions, ratios, decimals and percentages: Where two or three amounts are being compared and the value of some of the amounts, the difference between the amounts or the total is unknown. This can also be a very effective way to calculate selling price and cost price when given percentage profit/loss

 

T-charts

A T-chart is simply a table, usually divided into two columns, giving it a T-shape. They can be used as a means to aid calculations and/or to identify patterns and connections within problems .

Other strategies

Other strategies used in Operation Maths which promote the visualising and decoding of problems include:
• Using number bonds and branching
• Making lists
• Using “guess and test” (also known as Trial & Error)
• Using the process of elimination (e.g. logic problems)

You’ve been framed! A closer look at ten-frames

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Category : About Operation Maths

What is a ten-frame?

A ten-frame is a simply a rectangular frame, with 2 rows of 5 squares,  into which counters  or cubes can be placed to illustrate numbers less than or equal to ten. They are extremely useful resources to aid the development of number sense within, and beyond the context of ten. The use of ten-frames was developed by researchers such as Van de Walle (1988) and Bobis (1988).

They can help children:

  • keep track of counting
  • see number relationships eg odd and even numbers, doubles, near-doubles, number bonds
  • understand and learn the number bonds of numbers to and above 10
  • develop their understanding of place value
  • in their learning by being  part of a larger CPA approach to maths instruction

 

 What about a five-frame or  a twenty-frame?

While the ten-frame is the most common arrangement, multiples can be used to demonstrate numbers beyond ten eg 35 could be shown using three full ten-frames and five on a fourth frame. For exploring numbers up to five (eg with junior infants), a five-frame could be used; however, it is perfectly acceptable to use a ten frame and limit your use to just the numbers up to five (ie the top row).

The Operation Maths programme provides FREE frames with all the junior end books; five-frames for junior infants, ten-frames for senior infants and double-ten frames/twenty-frames for first and second classes. You can also show a digital version of the five-frame or ten-frame using the sorting eManipulative (see below) accessible on edcolearning.ie

 

Horizontal or vertical?

The most common configuration for a ten-frame is to use it five-wise (horizontally) and this is how they are shown in the Operation Maths books. However, the alternative pair-wise (vertically) configuration can also be used and both configurations have their merits:

  • The five wise (horizontal) configuration encourages links to the benchmark of five (see more on benchmarks below) and typically counters are laid out on the top row first, starting on the left ie 7 is 5 on the top and 2 on the bottom, therefore 5 + 2 = 7 (see image above)
  • The pair wise (vertical) configuration is very useful when emphasising the idea of doubles, near doubles, in-between doubles, odd/even numbers, halves etc. When using ten frames in this way, the counters are usually laid out on the bottom row first, starting on the left ie 7 is 2, 2, 2 and 1 on the left. The 100 square eManipulative, again accessible on edcolearning.ie can be very useful to show this configuration (choose the counters only option and then hide all counters, revealing only what is required)

I would encourage teachers to alternate between both layouts, as this encourages the children to develop flexibility in their thinking, which is a vital requirement in the attainment of mathematical fluency. Similarly, while it is advisable initially to stick to the traditional way of laying out counters/cubes as described above, when children are comfortable with those configurations they should then be encourage to identify the number of counters when arranged more randomly; for example below the children can be challenged to identify the number of counters below and to explain how they came to that answer.

 

Four relationships for number sense

Van de Walle lists four relationships that children should develop with numbers one through ten, all of which are ideal to be explored and reinforced using ten-frames:

  • spatial relationships
  • one and two more than/less than relationships
  • benchmarks of 5 and 10
  • part-part-whole relationships

 

 Spatial relationships and subitising

Spatial relationships is the ability to recognise an amount by its shape. Similar to subitising, which is the ability to identify a number of objects at a glance (ie without counting) the use of ten-frames encourages the simultaneous development of both these closely-related skills ie  if shown the standard horizontal configuration of seven the children might explain how they recognise it eg

  • “The top is full so that’s 5 and there’s 2 on the bottom so that’s 7”
  • “I see 3 empty spaces so it must be 7 because 7 and 3 is 10”

However, the children don’t need to start by instantly recognising a number in a frame, rather a progression might look like this:

  • Initially, without using of identifying amounts/numbers, the children are shown two different representations and asked to identify which has more/which has less.
  • The children can be asked to reproduce a pattern created by the teacher eg he/she shows a layout on a frame and children copy  this and show it on their own frames (no numbers)

Again the teacher should vary the representations: initially use five-wise (top row then bottom row) and pairwise (bottom two cells and up) configurations and then progress towards random arrangements, which are more challenging and allows the children to say what they see.

 

One and two more than/less than relationships

At this point, and within the specified number limits for the class, the teacher can show an amount on a frame eg 7 and then ask how many there would be if one more was added. The children should be encouraged to visualise this, suggest answers (eg they could write this on their Operation Maths MWBs) and explain their reasoning before using the counters/cubes and frames to confirm the answer. Initially, the children may have to count all the counters again, whereas ultimately, it is hoped that they will realise it is more efficient to count on.

Once comfortable with this, the process can be repeated to ask how many there would be if one counter was taken away (a simple introduction to subtraction as deduction), if two more counters were added and if two were taken away.

 

Benchmarks of 5 and 10

Through repetitive use of the ten frame, the children should already be developing an understanding of the numbers to combine to make these important benchmarks eg 7 + 3 = 10, 4 + 1 = 5 etc. The children can record the benchmarks using number sentences and/or branching number bonds (see opposite). Branching bonds are more visual and less abstract than number sentences alone as it is easier to visualise how 4 and 6 are combined to make 10 and they do not necessitate the use of operational symbols.

Other manipulatives such as the math rack/rekenrek (which is used in Mata sa Rang) also encourage children to think in terms of groups of fives and tens.

In first and second classes, the benchmarks should expand to include 20 and in higher classes other benchmarks, such as 100, are also important.

 

Part-part-whole relationships

Children need to appreciate that amounts/numbers can be broken down/decomposed into other amounts/numbers and that they can can also be combined to make larger amounts/numbers. In this way, the benchmarks of 5 and 10 are themselves examples of part-part-whole relationships but now the relationships should also include all the other numbers within the limits for the class.

Once children have grasped this understanding, they can begin to apply that to basic number facts (eg addition and subtraction) as they discover new strategies to arrive as answers without having to count all/count on. One of these key strategies is “Make 10” (see below) where the children change a less familiar fact into an easier fact by moving 1, 2 or 3 counters to make 10. Also known as compensation, this is a key strategy which can be applied to much larger numbers in higher classes. It also demonstrates the immense value of ten frame experiences in the junior classes and how they contribute towards the development of a child’s number sense that goes far beyond the less complex computations expected in the junior end classes.

Further reading:

Subitizing: What Is It? Why Teach It? By Douglas H. Clements

The Power of Subitising by Christina Tondevold, The Recovering Traditionalist

Building the benchmarks of 5 and 10 by Christina Tondevold, The Recovering Traditionalist

The Make 10 Strategy by Christina Tondevold, The Recovering Traditionalist

A Sense of ‘ten’ and Place Value from nrich.maths.org

What is a Ten Frame and why is it a useful tool for developing early number relationships and fact fluency?

Ten Frame Activities

Singapore Maths & Operation Maths

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Category : About Operation Maths

What is Singapore Maths and what has it got to do with Operation Maths?

When comparing international mathematical achievement at primary and secondary level, the Trends in International Mathematics and Science Study (TIMSS), is generally regarded as one the best comparison tools. And even a quick review of the score tables of these studies will highlight the consistent appearance of one particular country at the top – Singapore.

Singapore’s consistently high achievement has drawn attention and interest from educationalists internationally, keen to learn from the Singapore successes. And this has led to the buzz word “Singapore Maths” been given to both the maths curriculum and the way maths is taught in this country.

For the most part, the maths content in Singapore Maths is the same as the maths content in most countries, including Ireland. However, Singapore Maths is more than just content; primarily, it is a philosophy for mathematics instruction, in other words it’s more about how to teach maths than it is about what to teach.

In a similar way, the Operation Maths programme is significantly different to other maths programmes in the way it emphasises the importance of children understanding maths, and not just doing maths. Indeed, Operation Maths has been heavily influenced by some of the key elements of the Singapore Maths philosophy and many of  these feature strongly  in its own approaches.

Let’s look at some of the common elements of Singapore Maths and Operation Maths

Singapore Maths

Operation Maths
Demonstrates a concrete, pictorial, abstract (CPA) sequence of instruction based on the work of Jerome Bruner in the 1960’s Also based on a CPA approach, where the TRBs and pupils’ books illustrate how concrete materials can be used to model the concepts and, in particular, the more complex and abstract elements of primary maths in the middle and senior classes
Places huge emphasis on the base-ten system and how a solid understanding of place value will greatly enhance a child’s understanding of operations, decimals, measurement etc Also recognises the huge importance of base-ten understanding and has been specifically designed to allow more time for exploration of the place value concepts so as to give the children the best possible head-start on all the related concepts
Promotes the development of specific problem solving strategies (including bar models)  in a structured and developmental way Also enables the children to explore and use specific strategies throughout the classes and is the only programme currently that enables the children to understand and use bar modelling as a specific problem solving strategy
Encourages the development of mental computation skills via the use of various strategies to decompose and combine numbers to arrive at efficient and accurate answers. Emphasises the importance of flexibility over procedures Similarly, Operation Maths places a huge emphasise on key strategies such as doubles, number bonds and strategies for the basic number facts which encourage the children to become flexible thinkers.
Emphasises the importance of visual structures to illustrate concepts eg ten frames, number bonds, part-whole models and branching all help to illustrate the relationships between numbers and to help show how the numbers can be manipulated to solve calculations All of these strategies are also included in Operation Maths and in particular ten frames are included free with all the junior end books
Believes that everyone can experience success in maths so long as they are taught it correctly and that they also put in the effort to learn and persevere. Similarly, Operation Maths uses key learning statements (i.e. “I am learning to …”) which makes learning and success more attainable for all children
The pupils’ books present the content very visually and encourage the exploration and manipulation of concrete materials by the children Similarly the Operation Maths books have been designed to be very visual, and incorporate a whole, host of visual strategies, rather than relying on just digits, symbols and calculations, which can be too abstract, except for those more mathematically-able.

So there you have it…Operation Maths is like a taste of Singapore with a definite Irish twist!

The monthly topic in the junior classes

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Category : About Operation Maths

In the junior classes (i.e. junior infants to second class) the Operation Maths TRBs also list a monthly topic in the plans. This posts explains the rationale behind the monthly topics.

For each of the ten months of the school year, a topic around a particular maths concept is integrated with the planned curricular work eg see data  and 3D objects above. The purpose of the monthly topic is for the children to experience a strand unit in an informal way before deeper learning through a more formal approach.

As the integration of these topics takes place throughout the school year (see linkage above), the children’s knowledge in these vital areas is deepened and consolidated as a matter of daily routine. The relevant vocabulary is also gently introduced.

In the TRBs, there are suggestions of how each monthly topic might be incorporated (see below) and, of course, teachers can incorporate the topics further using their own activities.

This approach of using a monthly topic has been recommended by the PDST, in their junior infant scheme and senior infant scheme, where they state “Traditionally, other strand units such as ‘capacity’ would have been addressed in one two-week block, not allowing time for any number work. By exploring ‘capacity’ 2 days/week but extended over a 3 or 4 week-period, it is envisaged that pupils will retain their conceptual understanding of this strand unit, whilst Number and EMA conceptual development is on-going. Teachers using this planning approach have found that pupils are more likely to make connections between their Number work and the other strand unit.”

Operation Maths Digital – a completely integrated print and digital package!

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Category : About Operation Maths

Operation Maths provides an extensive range of digital resources with endless possibilities. In this post, I will discuss the various types of digital resources that are available and how they can be used in the classroom.

Overview

The Operation Maths digital resources include:

  • e-Manipulatives which can be used  as Ready to go activities and Create activities
  • Maths Around us videos
  • Write-hide-show videos
  • Scratch activities
  • Follow-on weblinks

 

e-Manipulatives

The fully flexible, easy-to-use, online e-Manipulatives are designed for teacher-led learning and to encourage whole-class participation. This impressive range of e-Manipulatives is optimised for use on an Interactive Whiteboard or a whiteboard with a projector so that teachers get the best results every time. They also facilitate a CPA approach to maths instruction.

The full range covers key maths areas:

  • Sorting & Shop e-Manipulative
  • Place Value e-Manipulative
  • 100 Square e-Manipulative
  • Bar Modelling e-Manipulative
  • Counting Stick e-Manipulative
  • Fractions e-Manipulative
  • Clock e-Manipulative

 

 

Lets look at each of these in more detail:

 The Sorting & Shop e-Manipulative allows the teacher to easily drag and drop shapes, animals, fruit, classroom objects, shop items, upper and lowercase letters, and numbers onto a workspace. It can be used blank or with various backgrounds, including frames, sets,  2×2, 5×5 grids etc . Of all the backgrounds, the shop background is particularly useful as it allows the teacher to create a shop scene with price tags, coins and sale tags, which can be used to explore a wide range of mathematical scenarios such as using small amounts of money in infants right up to scenarios involving percentage increase and decrease in the senior classes.

The Place Value e-Manipulative provides a wide range of place value tables which the teacher can use to demonstrate re-grouping. Each place value table contains either base-ten blocks, counters to represent the place value discs that accompany the 3rd-5th books, straws or money, and decimal values are included in a selection of the tables. Two tables may be shown on screen at the same time to facilitate comparisons between numbers. There is also the facility to display up to 5-digit whole numbers, which, in my experience, had not been possible previously as all other interactive manipulatives only extend to 4-digit numbers at most.

The 100 Square e-Manipulative is another extremely versatile tool. It can be used in numbers only, counters only or counters and numbers. You can very easily hide and reveal individual cells, whole sections of the grid or the entire grid.  I have gotten a huge amount of use of out it recently, with first and second classes, using it in numbers only mode, hiding all the numbers and just revealing one number. I then ask the children what number comes after/before this, what numbers is missing above/below etc. This is particularly good to assess the children’s ability to identify numbers around the decuples/decades (ie 30, 40, 50 etc) which are widely recognised as hurdles for many children.

This tool can also be used to model the 100 dots grid (on the inside back cover of Operation Maths 3 and 4) as a means to explore the commutative and distributive properties and the connections between various groups of facts.

The Bar Modelling e-Manipulative allows the teacher to create the bar models used in the text books quickly and easily. Bars can be dragged, dropped and resized and the teacher can change their colour. The teacher can also type and draw freehand on the workspace, making this a very useful resource for demonstrating the strategy of bar modelling

The Counting Stick e-Manipulative replicates the physical counting stick that a teacher might use in the classroom. The teacher can set the starting value and the steps value, and reveal or hide numbers along the counting stick. Decimal and negative numbers may also be shown on the Counting Stick e-Manipulative and two counting sticks can be shown at the same time, in order to compare various numbers.

The teacher can use the Fractions e-Manipulative to present fraction bars (linear models), fraction circles and pizzas (both area models). The teacher can change the fraction that is shown on screen, randomise fractions and hide or show the fraction
value, decimal value and percentage value. Two fractions may be shown on screen at the same time.

Analogue and digital clocks are provided with the Clock e-Manipulative. The teacher can choose to show one analogue clock, one digital clock, two analogue clocks, two digital clocks or an analogue and a digital clock at the same time.

 

All of the e-Manipulatives can be used as Ready to go or Create activities

Ready to go activities are already set up within each e-Manipulative with pre-programmed questions that appear on screen, meaning that the teacher doesn’t have to waste time looking in a book for the accompanying questions. The questions can also be answered on the children on their MWBs, thereby encouraging whole-class participation.

Create activities are so called because the teacher can open the e-Manipulatives and choose how to use it to best suit them, their class and the concept at hand. There are suggestions for Create activities printed in the TRB which show how the tools can be re-used in infinite ways to achieve a countless number of specific learning outcomes. And the Ready to go activities themselves will also provide the teachers with examples of how each e-Manipulative may be used.

Operation Maths videos

There are two types of videos; Maths Around Us videos and Write – Hide – Show videos. These videos have the advantage that they were custom-built to align with content in the children books and the commentator’s voice is noticeably Irish, which make them stand out from those video series that have been imported from other countries.

Another advantage of these is that they have been designed so the teacher only needs to press play and the questions and wait times are all built in, allowing the children to look, listen and responses on their MWBs. This means that they not only encourage active participation but they allow the teacher the opportunity to informally assess the pupils via their responses.

Maths Around Us videos

The series of Maths Around Us videos is full of real-world examples of maths in the environment and provides numerous opportunities for discussion and engagement. Take a look at this sample video below:

 

Write – Hide – Show videos

These are videos of the e-Manipulatives in use that focuses on the teaching method of ‘Write – Hide – Show’. These videos provide quick, easy-to-use scenarios and set-ups that engage children and pose meaningful maths questions. They also showcase the flexibility of the e-Manipulatives and provide inspiration for teachers’ own expansions. Take a look at this sample video below:

 

Scratch programming activities (3rd to 6th class)

Not only have these activities been written especially for Operation Maths but Operations Maths is only the only maths scheme available currently in Ireland with integrated programming (coding) activities. Each activity is integrated with the Pupils’ Books, comes with step-by-step instructions for teachers and pupils and highlights the connection between maths and coding in an easy-to-follow, visual manner.
Teachers or children can access the Scratch software for free online (click here).

Follow-on weblinks

Encourage your pupils to practice maths ideas at home with the useful Follow-on weblinks based on recommended games. Each Follow-on weblink is author-approved and is linked to a specific topic, for a specific class level, in the Pupils’ Book. The weblinks can be printed for children to take home and have fun practicing maths with their parents or guardians. Teachers can also use the weblinks in class as a lesson starter, for consolidation and assessment or, indeed, at any time.

And finally….

  • All the digital resources are all completely integrated with the print and eBooks; when viewing the eBook, the teacher need only click on the specific digital icon on the page to open the resource up in a new window/tab (ensure that pop-ups for the Edco Learning site are enabled)
  • Nearly all of the digital resources can be used in conjunction with the free mini white-boards, ensuring the maximum participation of the children.
  • As there are numerous ways to use each of the e-Manipulatives, they offer unlimited opportunities for assessment for learning and whole-class participation
  • They have been specially designed to help children to focus on the maths
  • They are user-friendly and approachable with bright, clear colours and layout

Teachers can access all the Operation Maths digital resources through Edco’s dynamic online digital hub, www.edcolearning.ie.

Enabling Computational Fluency

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Category : About Operation Maths

 

‘Fluency requires the children to be accurate, efficient and flexible.’ (Russell, 2000)

One of the main aims of Operation Maths, from junior infants to sixth class,  is to enable the children to be computationally fluent. And, the main way to achieve this, is to enable them to become flexible thinkers.

In the presentation below from the IPPN Education Expo 2017, the concept of computational fluency is explored, while also outlining how Operation Maths supports this approach in its programme.

Operation Maths: also the most child-friendly primary maths programme!

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Category : About Operation Maths

Not only is Operation Maths the most teacher-friendly programme currently available, but it is also the most child-friendly. Read on to find out why.

Enabling children to understand maths, not just do maths

In case you haven’t picked up on it already, Operation Maths is all about understanding, understanding, understanding!

As mentioned before, Operation Maths is based on a concrete, pictorial, abstract (or CPA) approach.

Concrete activities and experiences are emphasised and outlined throughout the pupil books. The Teachers Resource Book (TRB) also provides further suggestion for the ways in which a teacher can use concrete activities with their class.
These experiences are then further developed via the pictorial activities; this may be through interaction with the digital resources and/or via the write-in activities in the books. Of the CPA approach, exploring pictorial representations of materials, numbers, shapes etc., is a vital (but sometimes neglected, particularly at the senior end) step on a child’s journey towards understanding.
All of this should greatly enhance the children’s ability to visualise and understand maths and be more prepared when they meet the more traditional, abstract activities.

Active participation

The high number of concrete based activities within Operation Maths ensures that the children have regular, if not daily, opportunities to be engaged in active learning. The inclusion of free mini-whiteboards  (MWBs) as part of the programme also encourages the active involvement of all the class in any activities as all children must visibly participate. Moreover, the custom-made suite of write-hide-show videos and Maths around us videos (see below for samples) further increase the opportunities to use the MWBs. I challenge you to show the estimation video to your class and not have them all excited and active!

Mastery curriculum

Research suggests that changing chapters/topics every week, especially in senior end and especially the case of number topics, is not conducive to developing a strong understanding for the various topics and that preferable to this is a mastery curriculum approach, where the children are afforded a longer time to engage with, and ultimately master, the concepts. Therefore, in Operation Maths, these topics have a dedicated two weeks block, allowing the children to fully engage with the concepts before moving to a different topic, thereby increasing the likelihood for the children to master the content. So, if you look at the contents page of the Operation Maths books, you will notice that there are fewer chapters than usual but that many of these are “double” chapters, designed to be taught over ten school days, as opposed to five.

A better understanding of the operations

Similarly, research suggests that teaching opposite or inverse operations together, allows the children to better grasp the connections between the two, therefore promoting deeper conceptual understanding. So, in the senior end, addition and subtraction are taught together as part of the same chapter(s), as are multiplication and division. A similar approach is taken to the operations in first and second class once the children have been formally introduced to subtraction.

A seamless transition…

I was primarily involved in authoring all of the number and algebra chapters in the 3rd to 6th class books, while my co-authors Michael Browne and Siobhán Kelleher authored the chapters for measures, data, shape and space for the same classes. This means that, for the children, their experience of the content is very cohesive as it flows seamlessly from one class to the next.
Similarly the 5th and 6th class books were written in light of the NCCA Bridging documents and Project Maths at junior cycle in second level. In particular, extra attention was paid to the correct and accurate use of mathematical terminology which also helps ensure that the child’s transition from primary maths to second level maths is also as seamless as possible.

Other ways in which Operation Maths promotes the development of deeper connections and understanding include:

  • Linkage: the topics are all taught in a very connected way, with cross references being made where appropriate. In particular, measures are not just confined to their own chapters: they are taught in an integrated way across all suitable chapters and particularly across number 
  • Integration: The use of maths across other subjects is also emphasised regularly and particularly as part of the suggested activities in the TRBs. In the senior end there are also themed revision pages in the Pupil’s Book, where maths is explored through History, Geography, Sports etc.
  • The books have a clean, uncluttered design making them easier for children to focus on the concepts and not be distracted by unnecessary images
  • In the senior end, the Pupil’s Book also has clear and detailed teaching panels (easily identifiable by their yellow background) right throughout each chapter, to aid both the teachers and the children, and can be particularly useful if the children are working independently or in small groups.
  • To help make connections with the environment there are dedicated Maths Around Us videos in the digital resources, Maths Around Us activities in the Pupil’s Book and Maths Trails in the TRB photocopiables and in selected Discovery Books

Operation Maths: the most teacher-friendly primary maths programme

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Category : About Operation Maths

Resource BooksDesigned by a team of practically minded teachers, our aim was to make it as simple as possible for any teacher to use Operation Maths, while also reducing their planning and preparation work. This is why the content of the Operation Maths Teacher Resource Books (TRBs) is the most comprehensive currently available.

For the senior end:

Each topic dedicated section in the Teacher’s Resource Book (or TRB which is free to adopting schools) includes:

  • An overview of the topic that doubles as a short term plan and includes learning outcomes, vocabulary, resources, differentiation, assessment, linkage, integration etc., etc. Simply photocopy this and use it both as a plan and as a way to record progress through each topic. All of these plans are also available to download as MS Word documents from www.edcolearning.ie
  • A brief synopsis of potential difficulties that a class or individuals may have with the topic.
  • A day-by-day breakdown (see opposite) of suggestions to teach the topic that is so comprehensive, the like of which has not been seen before. This includes daily suggestions for oral and mental starters, concrete activities, questions, games as well as guidance for the suitable pages to use from the books.
  • Suggestions for further activities that will further enhance the children’s ability to understand the topic
  • Detailed instructions for how to use the digital resources specific to that topic.

 

Also included in the senior end TRBs:

  • Long term planning is in the introductory section of each TRB and not only is there a plan/grid with month-by-month suggestions of when to teach each topic for every class but, specifically for those teaching in multi-classes, there are versions of the plans that would suit a 3rd & 4th class combination and another for a 5th & 6th class combination.
  • A starters bank containing a whole suite of oral and mental maths activities.
  • A games bank with suggestions for simple, fun games, most of which are based on readily available resources such as dice, playing cards etc.
  • Reinforcement photocopiables (a minimum of one per chapter) for the lower attainers (LAs) which provide opportunities for extra practice for those requiring it.
  • Early finishers photocopiables (one per chapter) for the higher attainers (HAs) providing them with rich tasks to challenge them to think deeper and broader about the concept.
  • Dear Family Letters: these slips, which can be sent home in a homework journal or copy, explain to the family about the topic and suggest how they might support the child at home.
  • Target boards: designed for display on the class IWB, these are accompanied by questions specific to each class level and chapter.
  • A range of other photocopiables and cut-outs which can be used to further enhance the children’s understanding of the topic at hand.

For the junior end:

For the junior end , the TRB each section covers a specific month and each month has an overarching theme eg Capacity,  which then connects the other activities even if then are from another strand unit. Each month section is then subdivided into fortnights to aid planning and then subdivided further into weeks eg January week 3.

Each of the monthly sections includes:

  • Two fortnightly overviews of the topic(s) that doubles as a short term plan and includes learning outcomes, skills, vocabulary, resources, concrete materials,  differentiation, assessment, linkage, integration and home-school links. Simply photocopy this and use it both as a plan and as a way to record progress through each topic. All of these plans are also available to download as MS Word documents from www.edcolearning.ie
  • A weekly breakdown (see below) of suggestions to teach the topic that includes suggestions for warm-ups (starters), activities, questions, games as well as guidance for the suitable pages to use from the books.
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  • Suggestions for stations in the 1st & 2nd class books that will further enhance the children’s ability to understand the topic
  • Suggestions for Aistear links in the infants books (see doc above).
  • Detailed instructions for how to use the digital resources specific to that topic.

Also included in the junior end TRBs:

  • Long term planning is covered with the yearly scheme in the introductory section of each TRB and outlines the strand units within each month and cross references this to the relevant pages within the pupils books.
  • A reproduction of the curriculum objectives for the specific class level and details when each is covered during the year
  • Early finishers photocopiables for the higher attainers (HAs) providing them with rich tasks to challenge them to think deeper and broader about the concept.
  • A range of other photocopiables, games and cut-outs which can be used to further enhance the children’s understanding of the topic at hand.
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Understanding maths, not just doing maths…the Operation Maths way!

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Category : About Operation Maths

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.

 

Programme Components

For pupils in infants to second class:

  • At School BookOperation Maths Junior End
  • At Home Book
  • FREE Pupil Assessment Booklet
  • FREE Mini-white board
  • FREE Frames (five, ten or twenty frame)

 

 

For pupils in third class to sixth class:

  • Pupils boSenior Endok
  • Discovery book
  • FREE Pupil Assessment Booklet
  • FREE Mini-white board
  • FREE place value manipulative

 

 

For teachers of adopting schools:

  • Resource BooksFREE Teacher copies of all the relevant pupil resources
  • FREE Teachers Resource Book (TRB) which contains all necessary plans, teaching strategies, photocopiables, games, starters etc.
  • 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.

Furthermore, Operation Maths is the most teacher-friendly  and child-friendly programme currently available.

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