Category Archives: About Operation Maths

Fostering the development of correct mathematical language and terminology

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Category : About Operation Maths

Last Friday, I was working with a group of first class children who were completing some first grade activities on Splash Math, an American maths site. While, on the plus side, the activities on this site are very visual and promote a CPA approach to mathematical instruction, on the down side, the first grade in the US isn’t aligned exactly to the maths curriculum for first class in Ireland, and so we have regularly encountered activities that might have unfamiliar language and terminology.

This was one of those days. We were looking at 2D shapes in the geometry section when a child said quizzically to me, “I’m stuck, miss”. The question was “How many vertices has the shape (a circle): 1, 3, 0, 2?”. I asked the class could anybody remember, from the previous day, what vertices meant? A flurry of hands went up to tell me “corners” at which point the child had no difficulty identifying 0 as the correct answer. Then I asked the children to remind me of all the other American words to do with geometry that we had come across the previous day, which I then recorded on the board for the benefit of all the children (see image below).

It brought home to me how correct mathematical language and terminology is much more prevalent in the primary maths curricula and texts of other countries, and how it is often even introduced much earlier, when compared to Ireland. And, how much of a disservice we do to children in Ireland if we try to shield them from this language in primary school, only to have it all thrust at them in secondary, where some children might wonder if it is the same subject they are doing at all!

It also reminded me of an RSE inservice I attended years ago, which stressed the importance of the children being introduced to the correct terminology for the body parts, so they might be able to properly communicate and report any incidences that might occur. In a similar way, should we not introduce children to the correct mathematical terminology, so as to enable them to communicate their thinking more clearly and to explain the approaches they took and the strategies they used?

That is why Operation Maths has been written as a programme which does not shy away from the correct mathematical language and terminology, rather it specifically uses words like commutative, distributive, associative, dividend, product etc when explaining concepts. Furthermore, when introducing new terminology it is done via concrete and pictorial activities with the back-up of  a range of images that enable the children to not just know the word, but to be able to picture it also, and in that way to truly understand the concept it describes.

As can be seen from the example above, new terminology and language is typically introduced as part of the teaching panels (yellow-coloured sections) and is often in a blue bold font to highlight it as being new/significant. The new term is then explained in simpler words and using visual examples to reinforce its meaning for the children. Since it is envisaged that these teaching panels would be presented/mediated by the teacher, this ensures that the teacher can help explain the vocabulary and that the child is not meeting the new term  in a random section of text.

The questions/exercises for the children that follow these teaching panels have also been specifically chosen to help reinforce the new term and consolidate the concept that it entails. These typically incorporate the use of concrete materials or pictorials representations (as in the case of the 100 dots grid/sheet mentioned above) for further exploration and reinforcement.

With all new terminology, when met again, there is typically some supporting text to remind the child and/or revise the meaning. Furthermore, the child can always consult the colourful glossary at the back of his/her pupil’s book if necessary.

Some of the advantages of using correct mathematical terminology in primary mathematics:

Preparation for second level: The NCCA has published a number of Bridging materials for maths, which encourage continuity between mathematics in primary and post-primary schools. Included in these materials, there is a glossary of terminology that teachers of 5th and 6th classes are encouraged to incorporate, where possible, so that children will be better prepared for second level maths, thus easing the transition from primary. This terminology was deliberately included in the Operation Maths books for 5th and 6th. Furthermore, where useful, some terminology was also incorporated in a simpler way in the Operation Maths books for 3rd and 4th so as to make the introduction more gradual.

Number Sense & Number Talks: The buzz in maths education circles is all about developing number sense. One approach that is being encouraged to support this is to have regular Number Talks to encourage the children to communicate how they mentally solved a calculation and to explore and discuss the various strategies that could  be used. The promotion of the development of number sense is a key principle of the Operation Maths programme, from the use of frames in the junior classes, right up to the use of thinking strategies, bar models and other pictorial structures in the senior classes. Similarly, 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 this process. Ultimately however, this is all dependent on the children having a well developed range of mathematical terminology, by which they can clearly communicate and express their ideas and approaches.

Maths on the internet: Most of the maths we access on the net is american-based, be it You Tube videos, teaching sites, games, drill and practice sites. In the case of the latter, in many schools and homes, the children are encouraged to access teaching, drill and practice sites such as Khan Academy, Manga High, Splash Math, IXL.com etc to complement their core mathematical texts. As a result, Irish children will likely encounter, initially, terminology that is unfamiliar.  However, if they have encountered this terminology in their Operation Maths books, this will better prepare them for these sites. Indeed for those children and classes who have regularly accessed these non-Irish sites, they will probably have developed an understanding of this terminology already and its inclusion in Operation Maths will be unlikely to faze them at all.

 

Some FAQs:

Is this mathematical terminology in-line with the Irish Primary Mathematics Curriculum?

This is taken direct from the curriculum:
Third Class > Number > Operations >
The child should be enabled to explore, understand and apply the zero, commutative and distributive properties of multiplication.

Thus, not only is the terminology in-line with the curriculum, it raises the question how a child could previously have been enabled to “apply the commutative property” without being able to explain what he/she was doing and why, and furthermore how he/she could explain this without using the word “commutative” or “turn-around fact”?

Is is worth noting that the Teacher Guidelines, that accompanies the mathematics curriculum here in Ireland, includes a limited list of symbols, numerals, fractions and certain terminology for each class level (p. 70). However, other more generic terminology (eg product, factor, dividend etc) has not been categorised according to class levels, which contrasts with the curricula of other countries where specific terminology is typically specified for each year level/grade. Therefore, in writing Operation Maths, the authors categorised terminology into certain class levels based on evidence and practice in other countries.

Are the children expected to learn off and define this terminology? 

Of course not. In the same way as a teacher might use such terminology as simile, metaphor, alliteration etc to explain writing concepts in English, it is hoped that the teacher would use and reinforce specific terminology when appropriate, and in this way some of the children might also pick up this vocabulary and use it themselves when communicating their ideas. But it is not suggested or encouraged that these terms be drilled and “learnt off”.

We have a high number of children with dyslexia/English as a second language; should we avoid Operation Maths because of the language?

Actually, quite the opposite. While the teaching panels of Operation Maths may have more mathematical vocabulary that the competitor texts, they also have many more visual images that explain and demonstrate the concepts, and both the teaching panels and the exercises that follow are more concrete-based and pictorial in nature. This will in fact be better for children with limited language or language difficulties, as opposed to texts which are largely just digits and symbols, which themselves can be too abstract, particularly for senior classes. Plus, deliberately avoiding this language in primary only moves the issue on to becoming a bigger one when those children go to second level.
As mentioned previously, all of the Operation Maths programme is based on a CPA approach,  from the Pupils’ Book to the Discovery book, which is dominated by visual, rather than text, activities, to the free place value materials and frames, to the digital resources, eManipulatives and videos all of which place the emphasis on visual representations of content. This makes Operation Maths the most suitable programme for any child who is more of a visual learner.

Further suggestions, hints and tips:

Repetition, repetition, repetition! Whenever a new term is encountered don’t expect the children to know it,  understand it and use it straight away; research suggests that a child will typically need to encounter a word 15-20 times before they will start to use it. This is why it is important to use the term at every suitable opportunity and why in Operation Maths the term will be used repeatedly in various contexts to help this.

Use glossaries: As well as the Operation Maths glossary, use Jenny Eather’s, Maths Dictionary for Kids to look up new terminology and explore the visual and interactive activities that typically accompany each term. Another useful resource are the Math Vocabulary Cards from the Math Learning Centre, available to use online or download as a  free app. However, bear in mind that, while a definition in a glossary is useful, new terms must be also understood from meaningful examples and contexts relevant to the child.

Maths Word Wall: Whenever you encounter new terminology display it on your maths wall for future reference. This can be printed out vocabulary posters from the internet or small flash cards/A4 posters created by the children themselves. Aim to always include a pictorial representation and not just text. There are also lots of printable charts and posters available to download free from Jenny Eather’s, Maths Dictionary for Kids .

Start a personal maths dictionary: This allows children to keep a personal record of the vocabulary they encounter. Operation Maths users can use the vocabulary sections in the Discovery Book, where the children in 3rd and 4th must match the term to a definition and to an example. In Operation Maths 5, the children must provide the term to match the definition and, in Operation Maths 6, the children must provide the definition to match the term, as well as drawing an example in both cases. Thus the activities are getting slightly more difficult at each class level while continuing to emphasise the visual representations.

Use Number Talks: Through the regular use of Number Talks the children will begin to appreciate how having a good grasp of the correct mathematical language can help them explain their thinking in a more accurate and efficient way during number talks. Furthermore, he/she will realise that it is easier to understand the approach of a peer when they use terminology that he/she recognises and understands.

Make it fun: Play games such as matching games or “Just a Minute” word games.

Use matching activities, true or false, always, sometimes, never true etc: These type of language activities are included in the Operations Maths books to reinforce and consolidate the language acquisition. Also included are  oral discussion activities and “Talk Time” activities, to further promote discussion and exploration.

 

Further reading:

Developing mathematical vocabulary

Operation Maths 3-6: Managing the content

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Category : About Operation Maths

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

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Category : About Operation Maths

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.

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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

Tags :

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

Tags :

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

Tags :

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!

Tags :

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

Tags :

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.

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