# Monthly Archives: February 2017

## Operation Maths: A Unique Approach to Problem-Solving

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
• 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:

• 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

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

• 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

### 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.

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

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

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!

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

### 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.

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.