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):

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.

Maths by Month – September

It’s a new school year! And with it comes the first in a new series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant.

In each monthly overview, there will be links to other topic-specific posts and articles, as well as a whole host of extra suggestions, links etc. To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the blog via email, on the top right hand of this page.

Operation Maths Jr Infs to 2nd classes:

• Early Mathematical Activities (EMA) is the sole strand of Junior Infants for this month and the initial focus of Senior Infants. NRICH has a very comprehensive suite of activities and suggestions for early years, of which Socks, Packing, Collecting and Baskets are particularly relevant to this strand. The Ready Set Go Maths programme also has a whole suite of suitable activities.
• Number: counting & analysis of number in Senior Infants and Number: counting and numeration in 1st & 2nd classes both require similar skills, although the range of numbers will differ. Senior Infants will revise numbers to 5, 1st class will revise numbers to 10 and move on to the initial teen numbers, while 2nd class will revise numbers within 20 initially before expanding this to within 100. The Operation Maths frames are invaluable tools for this topic. Two informative blog-posts are Count on Me and Show Me. These one-minute videos from Origo Education, aimed at teachers, are also very useful:
  • Introducing the Counting Principles
  • Teaching the One-to-One Counting Principle 
  • Teaching the Stable-Order Counting Principle
  • Teaching the Cardinal Counting Principle
  • Teaching the Order-Irrelevance Counting Principle
  • Teaching the Abstraction Counting Principle
• Number: operations for First and Second classes will also be looking at addition and subtraction, within the limits of familiar numbers. The Operation Maths frames will once again prove invaluable here. Also recommended is this set of videos, again from Origo One, listed below. You may also notice that many of the strategies being demonstrated  in these videos are the same as those used in Number Talks (for more on Number Talks, please see the end of this page):
  • Introducing types of addition
  • Introducing types of subtraction
  • Exploring Add-to (active) addition
  • Teaching the Count-On Strategy for Addition
  • Teaching the Use-Doubles Strategy for Addition
  • Teaching the Make-Ten (or Bridge-to-Ten) Strategy for Addition
  • Teaching the Think-Addition Strategy for Subtraction
  • How should number facts be written?

• First and Second classes will also be looking at ordinal numbers (first, second, third etc) in Number: Comparing and Ordering. There is a list of online interactive games here and there are lots of useful videos on YouTube etc; just search for “ordinal numbers”.
• Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 
  1. Log into your edcolearning account
  2. Click on the At School Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.
• Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

Operation Maths 3rd to 6th classes:

Click on each link below to access more in-depth information and links on each of the topics for this month.

• Place Value (3rd-6th)
• Lines and Angles (3rd-6th)
• Representing and Interpreting Data (3rd & 4th)
• 2-D Shapes (5th & 6th)

Other suggestions for September:

• Check out the “Maths and me” attitudes questionnaire, situated after the last assessment in the Operation Maths Pupil Assessment booklet for 3rd to 6th classes. Suggest to the children that they fill this using a particular colour on one of the first days of the school year to be then revisited later in the year. At this point, the children can again record their attitudes in a different colour and reflect upon any changes they made, if any.
• Maths about me: another great activity for the start of a new school year. The children write facts about themselves that are appropriate to their ability eg height, age, shoe size, telephone number, distance from school (use google maps), time that they get up or go to bed etc. This can be recorded on the inside front cover of the discovery book, filled in on a pre-made template from the internet, used to make a large class display or even become a more complex problem solving activity in the more senior classes.
• Inspire your class for the year ahead: Most people have this belief that there is such a thing as a maths brain, a belief which Jo Boaler, among others, strongly challenges. In conjunction with her youcubed team at Stanford University, in 2015 they put together resources, videos etc for a Week of Inspirational Maths and followed that up with a Week of Inspirational Maths 2 and 3 in 2016 and 2017 respectively. The latter two have lessons and activities aimed at infants to 6th, as well as second level. Click on the link for an overview of the activities in Week of Inspirational Math, and scroll down to the bottom of the page to access all the resources; K-2 roughly align with Infants to 2nd and Grades 3-5 roughly align with 3rd-6th classes.
• New year, new initiative! Number Talks is an excellent maths methodology that is gaining traction globally, and more recently, nationally thanks to the promotion of the PDST. Better still, the rationale behind it aligns itself very closely with the underlying principle of Operation Maths, that is teaching children to understand maths, not just do maths. To find out more about number talks and to access a whole suite of ready-made resources for all class levels just click on the link above.

Maths for June

Hooray! June is here! You can almost smell the summer holidays!

And soon the testing will be all over (if it’s not already) and the books will be finished (if they’re not already). If you’re a user of Operation Maths 3-6 you are quite likely to be finished your books, as the programme is designed to be completed by the end of May, so as to have it all covered in advance of the standardised testing.

So now you might find yourself looking for inspiration to fill the maths lessons from now until the end of month. Whether you’re an Operation Maths user or not, look no further than the following ideas.

For Operation Maths users:

If you hadn’t had a chance to dip into these specific features of the Operation Maths programme so far this year then why not try these out now?

• Let’s Investigate! These sections are the last one or two pages at the end of the Pupils’ Books where the focus is on open-ended problems. Some of these are “big” enough to fill a whole lesson, others might become additions to a lesson or be combined to become a lesson. The children could also select which particular investigation(s) they’d like to explore either a whole class or with individual groups selecting different investigations, with results to to be communicated back to whole class when complete.
• Early Finishers Photocopiables: These can be found in your TRB and can also be  a great way to help deepen the children’s understanding of a topic covered earlier in the year. For 3rd to 6th classes, problem solving is also an integral part of these activities. In the TRBs for Junior Infants to 2nd classes, there are both Early Finishers photocopiables and dedicated problem-solving activities.
• Maths Around Us: If your class has access to recording devices, why not challenge them to make their own Maths Around Us video based on maths content they covered this year. Watch some of the OM Maths Around Us videos on www.edcolearning.ie for inspiration.
• As mentioned in a previous post, don’t feel under pressure to complete all of the above activities, only just what appeals most to you or is most suited to your class.

For everybody!

• Change their attitude to maths generally: Most people have this belief that there is such a thing as a maths brain, a belief which Jo Boaler, among others, strongly challenges. In conjunction with her youcubed team at Stanford University, in 2015 they put together resources, videos etc for a Week of Inspirational Maths and followed that up with a Week of Inspirational Maths 2 in 2016, the latter of which has lessons and activities aimed at infants to 6th, as well as second level. Click on the link for an overview of the activities in Week of Inspirational Math 2, which also includes links to all the required resources.
• Take time to problem-solve: often, during the school year, time is at a premium, yet Dan Finkel argues in this TEDx Talk that “allowing children time to struggle” is one of the Five Principles of Extraordinary Math Teaching. So after watching this video, why not present the images he uses to a 5th or 6th class and give them time to “notice and wonder”. The children could use sentence/questions stems like “I notice that…” and ” I wonder why/how/what ….” to get them thinking and discussing. Read on here for more sources of deep and rich problems.
• Try out a new methodology with your class. It can be a good idea to try out something new in June when there’s less pressure to succeed and you’re familiar with your class, rather than trying out something new in September when you’re trying to get to grips with new class, new books, perhaps new room etc!  One initiative I would wholeheartedly recommend is Number Talks. You could do a number talk with your class aimed at their current level or challenge them to do a number talks session aimed at the class they’ll be in next September.
• Do a maths project. In the Maths Curriculum Teacher Guidelines (DES, 1999) maths projects are listed as one of the examples of maths problems that we are encouraged to incorporate into our teaching. It can be difficult to include maths projects earlier in the year when the pressure is on to cover the content, making June an ideal time to explore them. For 10 “awesome” ideas, check out this post from the Mashup Math blog. I particularly like the idea of the child planning out their ideal holiday; so much real-life maths, costs, budgeting, estimating costs of luggage, time needed to get to the airport, distance from destination to airport etc. The NCETM Primary and Early Years Magazine also has suggestions for projects, the first one again focusing on financial education.
• Take it outdoors. Another type of maths problem listed in the Teacher Guidelines is maths trails. If the rain stays away for long enough why not get outside and do some maths trails? Or if you teach a more senior class, why not get them to design a maths trail for a junior class based on the school grounds or nearby environment. For more trail ideas read on here.
• Maths is Magic! There is a lot of mathematics behind magic. You could give the children magic tricks to investigate. Check out this article, again from the NCETM Primary and Early Years Magazine for sites to explore.
• Break the code: Explore the maths behind codes and code-breaking. You could ask the children to make up their own codes and crack a friend’s. Click here for links to suitable sites.
• Have a maths game-themed day. Another one of Dan Finkel’s Five Principles of Extraordinary Math Teaching is play. Most existing games and puzzles are mathematical. Get the children to bring in a game from home, to play in class, that requires mathematical thinking. Alternatively, get them to research a suitable one on the internet.

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?

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.

Further reading:

• Compensation is one of the strategies advocated for use by the PDST in their Teachers’ Manual Practical Approaches to Developing Mental Maths Strategies
• It is also one of the key strategies in Number Talks: Helping Children Build Mental Math and Computation Strategies by Sherry Parrish. Here, she lists it as suitable for grade 1-5 (US). Further information and slide presentations, based on the compensations as well as the other mental calculation strategies in her book, can all be accessed here

Fostering the development of correct mathematical language and terminology

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

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:

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
• 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)