# Operation Maths – Improving standardised test scores?

## Operation Maths – Improving standardised test scores?

Two days after the maths standardised tests were done in our school, a teacher on my staff came to me to let me know that 18 children in her room had gone up by a STen of 1 or more, a fact she was attributing to Operation Maths, which was being used throughout the school for the first time, since the previous September.

This information made me curious to see were there similar results in other class levels; below is a summary table of my findings:

Average Percentile for each class level, current and previous years in Drumcondra Primary Mathematics Test – Revised (DPMT-R):

 Current Class 2012 2013 2014 2015 2016 2017 Difference 2016 to 2017 2nd 55 61 +6 PR 3rd 63 61 77 +16 PR 4th 66 61 70 79 +9 PR 5th 63 61 58 68 68 +0 PR 6th 42 53 73 75 75 75 +0 PR

These results only include the classes that had a previous DPMT-R to which a comparison could be made. Also, they are the average of all the children’s results that completed the tests in each year, therefore other variables like children of different ability joining or leaving the school hasn’t been accounted for. However, they do make for interesting reading, while also raising interesting questions:

• 2nd, 3rd and 4th have made significant jumps, (3rd class in particular); could this be accredited to the Operation Maths programme (there were no other new initiatives this past year due to the freeze on the SIP for numeracy as directed by our union)?
• 5th and 6th classes stayed the same; why wasn’t the programme as effective for these classes? In the case of this school, perhaps the scores being already quite high in those classes meant there was little room for improvement. Or perhaps, because Operation Maths is a radically different programme, one that requires an openness to change the way we think about maths, it has more impact on younger classes where the children are more malleable and less rigid in their way of thinking than some of the older students. If this is the case, could we then expect to see improved results also for 5th and 6th class students in the future when they have been using the programme from 2nd and 3rd class?

Of course, this is only a small insight into one school’s experiences, and to have more conclusive results, data would need to be collected from a wider range of schools and this data would need to be monitored over time to see if these results were maintained. However, it does raise some interesting questions, and does indeed appear to indicate tentative evidence that Operation Maths can have a positive impact on the standardised test scores of all the children in a class. That said, improving test scores was never the main goal of Operation Maths, rather the aim is for the children to understand maths, not just do maths.  And if standardised test scores increase simultaneously, then that indeed is a positive bonus!

Did you use Operation Maths for the first time this year? Have you seen any similar trends with classes in your school? Please share with us!

Post script: Some may also suggest that Operation Maths has question items that better prepare the children for those in the test (i.e. teaching to the test). Having deliberately tried to be as unfamiliar as possible with the  DPMT-R test when authoring Operation Maths, means I can’t comment either way as I just don’t know if the question items resemble test items. Personally, I have no experience of the SigmaT at all, and at the time of authoring Operation Maths,  the only DPMT-R that I had administered in the previous 6-8 years was the DPMT-R for 5th class.

## Number Talks & Operation Maths

“The practice of number talks is one of the most powerful vehicles I know for helping students learn to reason with numbers and make mathematically convincing arguments, for building a solid foundation for algebraic reasoning, and for teaching mathematics as a sensemaking process. If all teachers make this shift in their practice, it would represent a profound advancement in mathematics education.”
Ruth Parker, co-author of  Making Number Talks Matter

As mentioned in a previous post, one of the mathematical pedagogies currently generating significant excitement is that of number talks. The buzz in maths education circles is all about developing number sense and number talks is being seen as one of the most powerful ways to enable this.

Here in Ireland, although the Professional Development Service for Teachers (PDST) has advocated the use of number talks in the PDST Mental Maths workshops and supporting manuals, and the more recent PDST Number Sense workshops, number talks is still relatively unknown. Similarly, there is very little in most of the maths text books available here, which explicitly promotes the development of specific mental maths strategies.

Not so Operation Maths. The promotion of the development of number sense is a key principle of the Operation Maths programme, as is the explicit exposure to a wide range of mental calculation strategies, most of which are also specified in the number talks approaches.

In this post, the connections between both number talks and Operations Maths will be shown, while also outlining how Operation Maths is the best programme to support the introduction and use of number talks in Irish classrooms. To read more about number talks generally, and access a whole suite of supporting resources  for all classes across the school,  please click here. To find out more about how Operation Maths works so well with number talks, please read on.

### What does a Number Talk look like?

One of the definitive number talks texts is Sherry Parrish’s book Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5. In this book, she recommends the following structure:

These actual number sentences or similar ones could be used as the basis for a number talks session (from Operation Maths 1)

### Other ways in which Operation Maths and Number Talks work so well together:

• In the junior end of the school, number talks is very much about the children developing their ability to conceptually subitise  (i.e. to recognise that there is 8 counters because there is a group of  5 and a group of 3) using a variety of images, including five and ten frames. Operation Maths also recognises the value of using frames throughout the programme in Junior Infants to Second class and provides these frames as part of the pupils’ book packs in these classes, as well as having digital eManipulatives  (i.e. the Sorting eManipulative) to support their use.
• In Sherry Parrish’s book Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5, she lists a whole range of specific strategies for the four operations, almost all of which are also explicitly taught or emphasized in the Operation Maths programme, including the strategy of compensation. To see an overview of the number talks strategies and where they overlap with Operation Maths click this link: Strategies in Number Talks & Operation Maths
• For those teachers using Operation Maths, they are already familiar with the structure of having an oral and mental starter at the beginning of each maths lesson. Number talks can be used interchangeable with the starters in the Operation Maths starters bank so as to add further variety to lessons.
• The strong emphasis on talk and discussion ( eg Talk Time in the pupils books, discussion and questions given in the TRBs) in Operation Maths further supports number talks as it prepares the children for situations in which they will be asked to explain their reasoning.

So there you have it, Number Talks & Operation Maths: a perfect partnership for each other!

## Are you compensating?!

A key recurrent theme in Operation Maths is the teaching of specific strategies to promote the development of flexible and fluent mathematical learners. In a similar way to the Building Bridges approach to reading, which advocates explicitly teaching specific reading comprehension skills, Operation Maths explicitly explores a range of specific strategies in a spiral and progressive way, in order to equip the children with the necessary skills for them to become capable and confident at problem-solving and computing mentally. Particular to mental computation, Operation Maths introduces the children to a range of of mental calculation skills, one of which is compensation.

### Compensation

Compensation is primarily an addition strategy where the aim is to to adjust one addend to become an easier number to add with.  This involves moving the quantity required to do this  from one addend to the other. In Operation Maths, these easier numbers are usually referred to as  friendly or compatible numbers and can include doubles, multiples of ten (10, 20, 30…) or, in the older classes, multiples of the powers of ten (100, 200, 300…..; 4,000,  5,000,  6,000 etc).

### Concrete

As with all new concepts and strategies, Operation Maths advocates a CPA approach. An ideal introduction to compensation is with the Operation Maths frames in first class when the children first begin to notice how adding onto 9 can be made easier by moving a counter from the other quantity to the 9 to make it become a ten. When ready, the children can also begin to explore how they can also make tens when adding to 8 and 7 by moving 2 and 3 counters respectively.

This can progress to using cubes  for bigger numbers; again, this should start with addends ending in 9 eg 19, 29, 39 etc. Encourage the children to see ways to make the calculations become easier, and encourage them to use the language of moving (not adding or subtracting) a cube from one number to the other, to make a friendly number. When ready, they should then develop this strategy to use with addends ending in 8 and 7, by moving 2 and 3 from the other number. In this way, the children can also begin to start doing addition with renaming, without having to grapple with the traditional written algorithm ( or column method).

### Pictorial

With first and second classes, it can be helpful also to show what is happening to the actual numbers in the calculation by using an arrow to highlight the quantity moving from one addend to the other. Notice how the calculation is being presented horizontally; this encourages children to consider the whole number and how it relates to the other number in the calculation. It also encourages the child to consider alternatives to the written column method, on which many children can be over-reliant.

In the senior end books for Operation Maths, branching (see red figures below) is used  to show the process of compensation and this can be particularly useful when the numbers involved are bigger than what might practically be shown using concrete materials. Never-the-less, it is always recommended to return to examples that can be demonstrated concretely, if the child finds the intermediary branching stage difficult to understand.

### Abstract

The ultimate aim is, that when presented with a random calculation, that the children will recognize and use compensation if it is an appropriate and efficient strategy. The suitability of compensation as an efficient strategy will depend on the numbers involved, which in turn requires flexibility on the child’s part. In most cases, this will only be likely, if they have previously encountered compensation, and a variety of other mental computation strategies, in structured  and meaningful lessons, like those provided by Operation Maths.

## Operation Maths 3-6: Managing the content

It can be very difficult to strike the correct balance of content in a maths programme;  a more able class might fly through the activities and conversely a less able group of children may work through content at a much slower pace. In a multi-class situation, the teacher may prefer to have more, rather than less, content so that one or more groups in the room can be kept occupied while the teacher is instructing a different group. Therefore, the volume of content required varies greatly from class to class and from school to school.

During the research and development phase of Operation Maths, the message from teachers was very clear: they wanted a maths programme with sufficient content and ideas, with no need to have to go sourcing extra material. Because of this feedback, the Operation Maths authors decided to err on the side of more, rather than less, content and designed a comprehensive maths programme that has considered everything a teacher may require, while also being able to be pared back to suits the needs of students and classes where a slower pace is preferable.

And not only is the Operation Maths programme highly adaptable to each unique teaching and learning situation, it is also based on the current, most forward-thinking approaches to maths education.

This post will provide some tips on how to best manage the programme in the senior classes, from third to sixth.

### 1. Start from the Teacher Resource Book

As always, when taking on any new programme it can sometimes take a while to discover the best ways to utilise it in order to maximise on its full potential for both you, as the teacher, and the children. Furthermore, since Operation Maths is based on many, very new and different approaches to the teaching of maths, this may leave teachers feeling a bit adrift initially.
That is why we recommend that those using Operation Maths for the first time should always start with the daily lesson suggestions in the Teachers Resource Book (TRB).  Typically, these will be laid out as follows:

• A recommended oral and mental starter, designed to consolidate prior learning and lead logically into the lesson that follows. It is suggested that this lasts for 5-10 minutes.
• The objective/learning  outcome for that day. This will also be given in the pupil book and/or discovery book
• Discuss and teach is the most important section. This will give suggestions on how to achieve the objective learning outcome. The suggestions will differ depending on the specific learning outcome(s); for example there may be probing questions given or suggestions for a concrete, pictorial or digital activity which may lead the children to deduce the new learning outcome(s) for themselves. It may involve reading and discussing a teaching panel (yellow panel) in the pupils’ book.
• Pupils’ book and/or discovery book: This gives the details for the location of the specific questions that reinforce and consolidate the learning outcome(s) covered in the discuss and teach section. It is not expected that all questions would be completed by all children and this is the main place where the teacher needs to decide what question activities are a priority for his/her pupils. Typically, the question sets are arranged to start with easier tasks and then graduate towards more difficult ones. There is often a section towards the end entitled “Work it Out” (blue panels in the pupils’ book) which contain the most difficult tasks and might be most suitable for the higher attainers (HAs) in the class.
• Digital Resources will list any relevant digital activities that can be used from the comprehensive suite on edcolearning.ie . These are given in the Pupils’ books as well and may also have been referenced previously in the discuss and teach section.
• Extra exploration: this is typically a suggestion of a game from the games bank that could be played by early finishers to reinforce the learning outcome of the day.

### 2. You don’t have to do it all!

As explained above, in the senior end TRBs , each topic is broken down into day-by-day plans which have a specific objective(s)/learning outcome(s) eg 5th class, Division, “I am learning to divide using chunking” or 3rd class, place value, “I am learning to identify the value of each digit in a number”. The discuss and teach sections lay out how to explore and teach each specific concept and what activities, either concrete, pictorial, digital or book-based can be used to reinforce the understanding.

However, it is not necessary that the class would do every single book-based exercise before they can move on. Rather, the teacher can select which ones they think most suitable for the ability level of their class. As explained previously, the initial question sets in each “day” are easier and then they progress in difficulty, often culminating in a  “Work it Out!” section. There are many different ways that a teacher could direct a class to answer these questions so as to facilitate differentiation:

• The children progress through the questions themselves at their own pace, as individuals or as pairs perhaps, for support.
• The children do a certain number in each question set eg first three in each; every second question etc
• The teacher could allow the children to choose what questions to answer eg ” I want you to do five questions, you pick which ones” or “I want you to select two questions out of each set, you choose”. The children could discuss at the end the reasoning behind their choices thus providing a great insight into their understanding of a topic and their concept of themselves as learners.
• The teacher could assign a number of incomplete questions as homework for that evening.

At the end of that “day”s maths lesson, it is likely that the children will have achieved the learning outcome, albeit to a variety of different depths, eg the child can identify the value of a digit in a number, even if not with 100% success rate. Irrespective of the content covered that day, in the next maths class, the teaching should move on to focus on the the next “day” and the next learning outcome(s), as envisaged in the day-by-day plans, thus ensuring that the children get a broad and balanced experience of the maths curriculum.

### 3. MWBs! MWBs! MWBs!

The free mini-whiteboards (MWBs) that accompany the Operation Maths programme are very adaptable  and can make covering content, in a meaningful way, so much more efficient. Some of the ways in which they can be used:

• Display  the ebook on your IWB (this is often preferable to the children looking at their own books  as they are looking straight up at the board, and therefore easier to check that the children are focused on the teacher and the task). Then, using  a selection of suitable questions from the book, you can use the MWBs for some fast-paced answering. This can be a great way to get through all/most of that day’s content, while also revealing any problematic questions/misconceptions that can then be focused on again as part of class-based reinforcement or practiced as part of homework.
• “Show your thinking”. The children can use quick jottings to explain how they arrived at a certain answer. The MWBs are less structured and easier to use than maths copies and are quicker and easier to change if you want to amend your ideas. Interesting responses or approaches could easily be brought up to the top of the class for further discussion and display.
• More maths done in less time. Rooting in bags, finding a copy, ruler, pencil, pen…ruling the copy, asking what date it is….this all leads to a delay in actually getting down to the maths at hand. Whereas, just writing on the MWBs is much quicker and gets more done.
• Bar models: This is one of the key problem-solving strategies used in Singapore Maths and a key strategy also in Operation Maths. If your pupils are not very familiar or comfortable with bar model drawing (for example if the children are using an Operation Maths book and didn’t have Operation Maths the previous year) it can be a great idea to draw the bar models step-by-step with the children i.e. the teacher draws on the classroom board and the children draw on their MWBs. Alternatively, the teacher can use the Bar Modelling eManipulative, available on Edco Learning to model the problems on the main IWB.
• Quick fire estimations: estimating should be quick responses and not take as long to produce as a full calculation would; otherwise they are not efficient (see this post for more on this). To practice these quick fire responses, you could quickly display a calculation on the class board from the Operation Maths ebook and then hide the calculation (eg use the no-show button on your projector remote) while the children quickly jot down estimated answers. These should then be compared and discussed, with reasons given as to why some estimates are more reasonable than others, before then agreeing on the most reasonable estimate(s).
• Step-by-step to show algorithms: if you are teaching some of the standard algorithms (eg the long division or long multiplication method) the MWBs can be handy to allow the teacher and class to do it together, step-by-step, with the children holding up their MWBs at every suitable juncture to check what they have done to that point. This way potential mistakes may be picked up quicker and addressed before they begin to occur repeatedly.

For other ideas on how you can use your Operation Maths MWBs across the curriculum, read on here.

### 4. Bar model drawing

As mentioned above, bar model drawing can be a difficult concept for both teachers and children to grasp when they’ve never come across them before. That said, they are an invaluable strategy and worth the investment; already feedback from teachers using the programme for the first time have revealed that topics the children previously found very problematic (eg fractions in all classes, cost price and selling price in 6th class), have now become so much easier and clearer, thanks to the structure of the bar models.

A way to make your collective introduction to bar models much easier, is to display the Thinking Blocks site on the class IWB and to get the children to respond by drawing the bar models and/or giving answers on their MWBs. Such an activity would also work well as an oral and mental starter that could be used regularly throughout the year.

### 5. Go digital!

The excellent suite of  digital resources available on Edco Learning can also aid efficient progress through content. The resources are very visual and help the child grasp a solid understanding of the concepts at hand quicker than might have occurred  otherwise. The resources can all be accessed directly via the hyperlinks in the digital books and it can be beneficial to have these tabs open in advance so as to save time during maths class. For more information on the extensive range of digital resources read on here

Teaching Junior Infants to 2nd class? Read on to find out how to manage the content for those classes.

## Why consider Operation Maths? Launch Presentation 2017

During March 2017, there were a number of Operation Maths launches around the country; Dublin West and North, Cork, Limerick, Kilkenny, Galway and Meath. At each of these launches the teachers were very interested in the Operation Maths programme and very impressed by all it has to offer.

Perhaps you were at one of these launches and would like a reminder of the various features of the programme explored on the night?

Perhaps you would like to share this information with colleagues and fellow staff?

Or perhaps you missed the launches and would like to find out more about Operation Maths for yourself?

If you answered yes to any of these questions, then below is a pdf of the slides from these launch nights. Please note that these have been compressed into a pdf to enable them to be uploaded, and so do not include the digital resources videos shown on the night. However, these can be viewed on this blog, under the tab Digital Resources

If you have any other queries about the programme, or would like samples, a demo of the digital resources, etc., please, don’t hesitate to contact your local Edco rep.

## A CPA approach to 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

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.”