Thinking Strategies for Multiplication and Division Number Facts

Thinking Strategies for Multiplication and Division Number Facts

What are number facts?

Number facts are the basic number facts that, it is hoped, children could recall instantly, so as to improve their ability to compute mentally and use written algorithms. Traditionally referred to as tables, the multiplication and division number facts typically include all the multiplication facts up to 10 x 10 and their inverse division sentences.

Some of the big ideas about number facts:

  • Some facts are easier than others to recall – which ones, do you think?
  • The easier facts can be used to calculate other facts – which ones, do you think?
  • The same fact can be calculated using various approaches – these approaches are often referred to as thinking strategies – see more below.
  • Using thinking strategies means that the children can apply the understanding, to facts beyond the traditional limits of “tables”.

 

What are thinking strategies?

A thinking strategy is a way to think about a process to arrive efficiently at an answer. For example, if asked to multiply a number by 2, one could double the number. Doubling is a very effective thinking strategy for the multiplication facts of 2, 4 and 8, as can be seen in the video below.

 

Halving is the opposite to doubling. And halving is a very effective thinking strategy to use for the multiplication facts of 5; if asked to multiply a number by 5, one could think of 10 times the number and then halve that amount (see below).

The Operation Maths  and Number Facts books for third and fourth classes repeatedly emphasise (among other thinking strategies) the strategy of doubling and halving known facts to derive unknown facts, eg through doubling I can work out 2 times, 4 times and 8 times a number; if I know 10 times the number I can work out 5 times, etc. 

From Operation Maths 3, possible thinking strategies for 2x, 5x, 10x.

The 100 dots grids on the inside back covers of Operation Maths 3 and 4 and Number Facts 3 and 4 can be extremely useful for the pupils to model various arrangements/arrays, while the teacher can use the Operation Maths 100 square eManipulative to replicate (and label) the children’s arrangements on the IWB.

Using doubling to model 2 x 6, “2 rows of 6”, 4 x 6, 8 x 6 (left) and trebling to model 3 x 7, 6 x 7, 9 x 7 (right)

Furthermore, multiplication and division are taught together throughout the Operation Maths series, so that, rather than compartmentalising each operation, the children develop a better understanding of how both concepts relate to each other. In this way, the basic division facts are easier to acquire, as they are understood to be the inverse of the more familiar multiplication facts. However, it is important that within each group of facts, the children explore the multiplication facts first; the better their understanding of these, the more likely they are understand the inverse division facts. Indeed, “think multiplication” is in itself, a thinking strategy for the division facts (see video below).

 

Traditionally, learning “tables” had been by rote, but current research suggests that this is ineffective for the majority of children. In contrast, children should be taught to visualise numbers and to use concrete materials, images and thinking strategies to use what they know to solve what they do not know. Below are examples of some useful thinking strategies for the basic multiplication and division facts (taken from Number Facts 3 & 4, Edco, 2018)

There can often be different ways to think about the same fact (or groups of facts), and the children should always be encouraged both to identify alternative approaches and to choose their preferred strategy. For example, consider 5 x 9:

5 times is half of 10 times: 10 × 9 = 90, so 5 × 9 = half of 90 = 45
9 times is one set less than 10 times: 10 × 5 = 50, so 9 × 5 = 50 − 5 = 45
9 times is treble 3 times: 3 × 5 = 15, so 9 × 5 = treble 15 = 45

Once the children understand how to arrive at an answer via a thinking strategy, they can then apply this thinking strategy to more complex calculations that are beyond the traditional 10 x 10 ceiling of “tables”; for example if I understand 5 times any number is half 10 times the number, then I can use this to mentally calculate 5 x 18, 5 x 26 etc (see more on this below).

 

Computational Fluency:

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

The primary aim of both the Operation Maths and Number Facts series (see more information on Number Facts below) is to enable the children to become computationally fluent. To achieve computational fluency, the children must be accurate, efficient and flexible:

  • Accurate: the children must arrive at the correct answer, e.g. 6 x 8  =48.
  • Efficient: the children must calculate the answer in an efficiently. A child who produces an answer of 48 in response to the question 6 × 8 by counting in jumps of six or eight may be accurate but is not efficient.
  • Flexible: children must be able to visualise and mentally manipulate numbers in order to see how they might be broken down and recombined to get an accurate and efficient answer (as shown with the various ways to consider 6 x 8 below).

Thus, flexibility is the key to fluency. A child who only knows that 6 x 8 = 48 becasue they have memorized that fact, is missing out on all the various possible connections between those numbers, subsequently hampering future connection-building. In contrast, a child who is flexible with number facts is one with a well-developed number sense, who can see the connections both between and within numbers, i.e. they can partition and/or combine numbers into more compatible (friendly) amounts and can apply their strategies to numbers beyond those they have dealt with. Therefore, a thinking strategies approach will not only be effective for aiding understanding and recall of the basic facts up to 10 x 10, a thinking strategies approach can enable children to apply these mental computation skills to numbers beyond this traditional ceiling, as shown below.

From Number Facts 4

 

The Number Facts Series from Edco

Number Facts is latest addition to the Edco Primary Maths stable, and it is a series of activity books designed to foster fluency in number facts for primary school children from First Class. The series features an innovative approach to the acquisition of basic number facts, and, like Operation Maths, teaches children to understand, not just do, maths.

Image result for number facts edco

In contrast to the more traditional drill-and-practice workbooks, which just test whether the answer is known, Number Facts teaches children to visualise numbers pictorially and to use these images and thinking strategies to become more adept at manipulating numbers. The specific focus of Number Facts will be to develop children’s thinking strategies and apply these to the basic number facts in such a way as to promote the child’s ability to visualise and recall these facts, thereby achieving fluency.

Both this rationale, and the suggested teaching approaches to the teaching of the basic multiplication and division facts for third and fourth classes, are clearly outlined in the Teachers Resource Book (TRB) which accompanies the series, and which is downloadable here. This TRB also includes a Long Term Plan for both third and fourth classes (see extract below), outlining a logical progression for the various fact groups throughout the school year. To view sample pages from the pupils Number Facts books please click here. Sample copies of all the books are also available from your local Edco reps.

 

Further reading and viewing:

 

 


Thinking Strategies for Addition and Subtraction Number Facts

What are number facts?

Number facts are the basic number facts that, it is hoped, children could recall instantly, so as to improve their ability to compute mentally and use written algorithms. Traditionally referred to as tables, the addition and subtraction number facts typically include all the addition facts up to 10 + 10 and their inverse subtraction sentences.

Some of the big ideas about number facts:

  • Some facts are easier than others to recall – which ones, do you think?
  • The easier facts can be used to calculate other facts – which ones, do you think?
  • The same fact can be calculated using various approaches – these approaches are often referred to as thinking strategies – see more below.
  • Using thinking strategies means that the children can apply the understanding, to facts beyond the traditional limits of “tables”.

What are thinking strategies?

A thinking strategy is a way to think about a process to arrive efficiently at an answer. For example, if asked to add 9 to a number, one could think of moving 1 from the other addend to the 9 so as to make a 10, which therefore becomes an easier calculation (see below)

      

The Operation Maths books for first and second classes emphasise three specific thinking strategies throughout: counting on from the biggest number, using doubles and near doubles and using the number bonds for ten (see image below). The doubles facts and bonds of ten are also included on the pull-out flap at the back cover to the pupils books, both for quick reference and to emphasise their importance.

From Operation Maths 2 At School Book

In the case of doubles, near doubles and bonds of ten, these key sets of number facts tend to be easier for children to understand and recall. These facts also make up a core section of the total addition facts to 10 + 10, as highlighted below on the addition square. When these become known facts, they can then in turn be used to calculate unknown facts (eg if 7 + 3  = 10, then 7 + 4 = 11), thus covering an even greater number of the total addition facts.

Furthermore, addition and subtraction are taught together throughout the Operation Maths series, so that, rather than compartmentalising each operation, the children develop a better understanding of how both concepts relate to each other. In this way, the basic subtraction facts are easier to acquire, as they are understood to be the inverse of the more familiar addition facts.

Traditionally, learning “tables” had been by rote, but current research suggests that this is ineffective for the majority of children. In contrast, children should be taught to visualise numbers and to use concrete materials, images and thinking strategies to use what they know to solve what they do not know. Below are examples of some useful thinking strategies for the basic addition and subtraction facts (taken from Number Facts 1 & 2, Edco, 2018)

From Number Facts 1 & 2

There can often be different ways to think about the same fact (or groups of facts), and the children should always be encouraged both to identify alternative approaches and to choose their preferred strategy. For example:

8 + 6 = (5 + 3) + (5 + 1) = 10 + 4 (make a ten) = 14
8 + 6 = 10 + 4 (move 2 from 6 to 8 to make a ten) = 14
8 + 6 = 7 + 7 (move 1 from 8 to 6 to make a double) = 14

Once the children understand how to arrive at an answer via a thinking strategy, they can then apply this thinking strategy to more complex calculations that are beyond the traditional 10 + 10 ceiling of “tables”; for example if I understand different ways to calculate that 8 + 6 = 14, then I can use these ways to mentally calculate 18 + 6 , 18 + 16 etc.

Computational Fluency:

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

The primary aim of both the Operation Maths and Number Facts series (see more information on Number Facts below) is to enable the children to become computationally fluent. To achieve computational fluency, the children must be accurate, efficient and flexible:

  • Accurate: the children must arrive at the correct answer, e.g. 8 + 6 = 14.
  • Efficient: the children must calculate the answer in an efficiently. A child who produces an answer of 14 in response to the question 8 + 6 by ‘counting all’ (eg have to count up to a total using using counters, fingers, etc.) may be accurate but is not efficient.
  • Flexible: children must be able to visualise and mentally manipulate numbers in order to see how they might be broken down and recombined to get an accurate and efficient answer (as shown with the various ways to consider 8 + 6 above).

Thus, flexibility is the key to fluency. A child who only knows that 8 + 6 = 14 becasue they have memorized that fact, is missing out on all the various possible connections between those numbers, subsequently hampering future connection-building. In contrast, a child who is flexible with number facts is one with a well-developed number sense, who can see the connections both between and within numbers, i.e. they can partition and/or combine numbers into more compatible (friendly) amounts and can apply their strategies to numbers beyond those they have dealt with. Thus, a thinking strategies approach will not only be effective for aiding understanding and recall of the basic facts up to 10 + 10, a thinking strategies approach can enable children to apply these mental computation skills to numbers beyond this traditional ceiling e.g. 19+ 5, 29 + 6 etc (see below).

       

The Number Facts Series from Edco

Number Facts is latest addition to the Edco Primary Maths stable, and it is a series of activity books designed to foster fluency in number facts for primary school children from First Class. The series features an innovative approach to the acquisition of basic number facts, and, like Operation Maths, teaches children to understand, not just do, maths.

Image result for number facts edco

In contrast to the more traditional drill-and-practice workbooks, which just test whether the answer is known, Number Facts teaches children to visualise numbers pictorially and to use these images and thinking strategies to become more adept at manipulating numbers. The specific focus of Number Facts will be to develop children’s thinking strategies and apply these to the basic number facts in such a way as to promote the child’s ability to visualise and recall these facts, thereby achieving fluency.

Both this rationale, and the suggested teaching approaches to the teaching of the basic addition and subtractions facts for first and second classes, are clearly outlined in the Teachers Resource Book (TRB) which accompanies the series, and which is downloadable here. This TRB also includes a Long Term Plan for both first and second classes (see extract below), outlining a logical progression for the various fact groups throughout the school year. To view sample pages from the pupils Number Facts books please click here. Sample copies of all the books are also available from your local Edco reps.

Further reading and viewing:

  • Are you compensating? A closer look at the thinking strategy of compensation.
  • Number Talks : this is a maths methodology centered around the development of  strategies and mental calculation skills. As such, it really complements both the Operation Maths and Number Facts series. For more information on where Operation Maths and Number Talks overlap, please read on here.
  • Mental Maths handbook for Addition and Subtraction from the PDST
  • Number Facts Board on Pinterest
  • The Origo One videos below are a great way to get an overview of some various thinking strategies, each in 60 seconds or less!

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of addition and subtraction, please check out the following post: Dear Family, your Operation Maths Guide to Addition and Subtraction


Start as you mean to go on!

Tús maith leath na hoibre!

So it is with every maths lesson. It is recommended that each maths lesson should start with an oral and mental starter, which:

  • reinforces some previous learning; not only does this serve to consolidate understanding but, if the content is more familiar to the child, this builds confidence and encourages participation.
  • should be active so as to further encourage the participation of all children eg using activities that incorporate mini-white boards (MWBs) requires more children to be involved
  • should only last for about 5-15 minutes; it should not take over the main part of the lesson

Below are some suggestions for oral and mental starters, both for those who are Operation Maths users and for those who are not.

Operation Maths starters:

In the Teachers Resource Books there are recommended oral and mental starters, designed to consolidate prior learning and lead logically into the lesson that follows. It is suggested that this phase of the lesson lasts for 5-15 minutes.

  • In the junior end TRBs for infants to 2nd class, within the weekly breakdown of suggested activities to teach the topic, there are suggestions for whole class warm-ups  and oral activities (starters).
  • In the senior end TRBs for 3rd to 6th class, within the day-by-day breakdown for each lesson there is an oral and mental starter listed (see image below); this is then explained in more detail within the starters bank, a section of the TRB that follows the topic chapters. To view a sample, click on the link to download the Operation Maths 5 Starters Bank

HINT: While there are typically many suggestions given in the Operation Maths TRBs, it is not necessary to do all of them. If you find a starter that works particularly well, you could note this alongside the margin of your TRB, or in the notes section, to highlight it for future use. And, if you are working with more than one class (ie multi-class), use the starter suggestions from the class level that suits the ability of the majority of the room. 

Other Starters:

There are many other types of starter activities that can be used interchangeably with the starters in the Operation Maths TRBs so as to add further variety to lessons.

    • Number Talks (infants to sixth) is an excellent maths methodology,which promotes the development of number sense and mental calculation skills. The rationale behind Number Talks aligns itself very closely with the underlying principle of Operation Maths, i.e.  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. To find out more about the overlaps between Number Talks and Operation Maths please read on here.
    • Same but different Math (infants to sixth) is a collection of fantastic images, arranged, in a very teacher-friendly way, according to topic. The teacher can pick out images relevant to the current topic, and suitable for the ability of the children and then ask them to come up with ways in which they are the same and also different. The children could use their MWBs in landscape layout, with a line drawn down the middle, on which to record points. Similar to this is Same or Different images
    • Splat! (first to sixth) from Steve Wyborney, is an engaging activity that helps build students number sense, while having math conversations. The difficulty increases from number bonds of ten through to multi variable equations. There is even a Fraction Splat! series. He also shares lots of free resources to aid implementation. Furthermore, a teacher could develop Splat! into a game/activity played in pairs or small groups, using concrete materials, where a child hides a number or quantity of objects/counters under Splats! (cut out pieces of card or fabric) for others in the groups to identify.
    • The Estimation Clipboard, (first to sixth) again from Steve Wyborney, encourages the children to look closely each time at set of four images, and to use what they have learnt from the initial images to refine their estimate for the latter images. Another number sense building activity on his site is Primary Tiles.
    • WODB (which one doesn’t belong), is based on four images/symbols/quantities, to which the children must give a reason for why one of them doesn’t belong. However, the content of the images has been deliberately chosen so that it could be argued that each one of the images doesn’t belong to the group! In this way, it encourages the children to think outside the box and appreciate that there is often more than one correct answer.
  • Thinking of a Number  (first to sixth class) is a simple but effective game to play with the whole class on the IWB as a starter. This is one possible way to use it:
      • Choose a number range that suits your class and click on three clouds to reveal their clues.
      • Ask the children (in pairs perhaps) to record all of the possible answers  on MWBs which are then revealed when called upon.
      • The children should look around the room to see if there are any possible answers given with which they do not agree (eg an even number written when one of the clues is that it’s an odd number) and to explain why they don’t agree.
      • Click on a fourth cloud to reveal the fourth clue; the children should X out all of the previous answers that can now be discarded and could be asked to explain why this is so.
      • Reveal the fifth clue; this should conclusively point at one actual answer. Again the children could be asked to explain why this is so.
      • On occasion, the actual answer may have already been identified by the fourth clue. In this case, ask the children to suggest what the fifth clue might be.
    • While Thinking of a Number is limited to whole numbers up to 100, once the children get the hang of the game they could be prompted to come up with five similar clues for a shape, measurement, fraction, decimal number etc. For more ideas on how to use this please check out this post here.
  • Bar Models are one of the three visual strategies for problem solving that are used and developed throughout the Operation Maths books for the senior end. While the children and the teacher are still less familiar with bar models, a great way to make your collective introduction to bar models much easier, is to use the Thinking Blocks site (which are based on bar models; suits second to sixth class) as an oral and mental starter. The teacher can 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.
  • Solve Me Mobiles are a fantastic suite of progressive puzzles that work as a lead-in to solving simple equations and variables in algebra. That said, these could be used from third class up (and perhaps even with  pupils in second class). Again, this tool will work well displayed on a class board and in conjunction with the pupil’s own MWBs. It also has the added advantage that the children can log-in  and use this site on a device at school or at home, so that their progress can be recorded and continued each time, rather than having to start from the beginning. Indeed, it would work well if the teacher sets up a generic account so that, even when using this with the whole class, they can pick up from where they left off.
  • That Quiz is an excellent assessment tool; it can also be used to generate a random selection of quick questions to which the children respond on their MWBs.
  • Operation Maths also includes useful Follow-on weblinks. Each follow-on weblink is author-approved and is linked to a specific topic and for a specific class level. As many of these are games, they could be used as a whole class starter (as well as for for consolidation and assessment) when displayed on the class IWB. The weblinks can also be printed for the children to take home and have fun practicing maths with their parents or guardians.

And if you exhaust all the ideas above there are some more suggestions on this list of Daily Routines and on this list of Useful Websites


Digging Deeper into … Addition and Subtraction (3rd to 6th class)

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of addition and subtraction, please check out the following post: Dear Family, your Operation Maths Guide to Addition and Subtraction

Addition and Subtraction is always the first operation’s chapter in Operation Maths 3-6, and it is always a double chapter i.e. it is structured to be covered over 10 days/two school weeks. In Operation Maths 3-5 there is also a second Addition and Subtraction chapter (this time only a single i.e. one week chapter) in the second half of the school year to revise and re-focus on specific strategies that can be used.

Relationship between addition and subtraction

In contrast to traditional maths schemes, which often have separate chapters for each operation, Operation Maths instead teaches addition and subtraction together, as related concepts. Teaching the operations in this way will encourage the children to begin to recognise the relationships between addition and subtraction, and indeed all the operations. Thus, the initial activities in the Discovery Book, require the children to reflect on their understanding of the concepts and to compare and contrast them.

In particular, the children are enabled to understand addition and subtraction as being the inverse of each other and are encouraged to use the inverse operation to check calculations.

Looking at the bigger picture

Children can often have tunnel vision (or column vision) regarding addition and subtraction calculations: they “do” the units, then the tens, then the hundreds without really looking at the whole numbers or the processes involved.

One way in which you can encourage the children to look at and understand these operations better is by using a Concrete–pictorial–abstract (CPA) approach. This means the children will be moving from experiences with the familiar base ten concrete materials (e.g. straws, base ten blocks, money, the Operation Maths place value discs, pictured above) to pictorial activities (e.g. where the children draw representations of the numbers using pictures of the concrete materials or use empty number lines, bar models, etc.) and finally to abstract exercises, where the focus is primarily on numbers and/or digits.

When exchanging tens and units or tens and hundreds, reinforce that a ten is also the same as 10 units, and that a hundred is the same as 10 tens and 100 units.
The use of non-canonical arrangements of numbers (e.g. representing 245 as 2H 3T 15U or 1H 14T 5U), as mentioned in Place Value, can also be very useful to children as they develop their ability to visualise the regrouping/renaming process. The Operation Maths Place Value eManipulative, accessible on edcolearning.ie,  is an excellent way to illustrate this and explore the operations in a visual way. Worth noting also, is that the Operation Maths Place Value eManipulative and place value discs provide the only means to concretely or pictorially represent base ten materials to five whole number places (no other interactive tool is available on the internet to do this); a fact which will be of particular value to teachers of 5th and 6th classes who didn’t have a way to concretely/visually represent numbers to ten thousands prior to the inception of Operation Maths.

Mental strategies are as important as written methods

The traditional, written algorithms for addition and subtraction, i.e. the column methods, are important aspects of these operations. However, in real-life maths, mental calculations are often more relevant than written methods. Also, as mentioned previously, children can often have tunnel vision (or column vision) regarding addition and subtraction calculations; they ‘do’ the units, then the tens, then the hundreds, etc., without really looking at the entire numbers or the processes involved. Therefore, while the column method for addition and subtraction is a main part of this topic, equally important is the development of mental calculation skills, using such strategies as those outlined on this page from Operation Maths 6 (below)

Thus, one of the main purposes of the Addition and Subtraction chapters in Operation Maths is to extend the range of mental calculation strategies the children have and to enable them to apply the strategies to numbers of greater complexity i.e. for the children to become efficient and flexible, as well as accurate. As the same calculation can often be done mentally in many different ways, the children have to develop their decision-making skills so as to be in a position to decide what is the most efficient strategy to use in each situation.

It is worth noting that the page from Operation Maths 6 pictured above serves as a synopsis to remind the children of all the strategies they explored individually in the previous Operation Maths books. That said, if the sixth class children are new to Operation Maths and have never encountered these strategies before, they may need to progress at a much slower pace than those who have been using the programme previously, or who may have encountered these strategies, for example a class who used Number Talks. As mentioned in a previous post, the Operation Maths mental strategies listed below are very similar to, and in some cases identical to, those used in Number Talks (if different terminology from Operation Maths is used in Number Talks, the Number Talks terminology is given in brackets).

  • Doubles and near doubles
  • Number bonds of 10, 100 and 1,000 (Making tens)
  • Friendly or Compatible numbers (benchmark/friendly numbers)
  • Partitioning (breaking each number into its place value parts)
  • Compensation
  • Adding up in stages/sequencing (adding up in chunks)
  • Subtraction as take-away (removal/deducation)
  • Subtraction as difference (adding up/complementary addition)
  • Constant difference subtraction (see below)

Operation Maths also places particular emphasis on the development of estimation skills for number and introduces and develops specific estimation strategies as the books progress. Again, the emphasis is on the children contrasting and comparing these strategies and choosing the most efficient strategy each time. To find out more about some of the estimation strategies, read this post.

Therefore, ask the children, as often as possible when meeting new calculations, can they do it mentally, and how, so that they become increasingly aware of a range of mental calculation skills and approaches. In this way the children will also be developing their decision-making skills, so as to be in a position to decide the most efficient strategy/approach to use.

Problem-solving strategies

One of the main aims of Operation Maths 3-6 was to introduce both teachers and pupils to a logical problem solving approach (i.e. RUCSAC) , complemented by specific visual problem solving strategies which develop in complexity as the child progresses through the senior classes.

A key step in the RUCSAC problem-solving approach is the ability to read a word problem meaningfully, and highlight the specific operational language or vocabulary. This is reinforced with activities in the Discovery Book (see below) where the children colour-code the specific phrases and then transfer them to their Operations Vocabulary page towards the end of their Discovery Book for future reference.

You will notice that the problems have no numbers to distract the children, so that they can just focus on the language of the problems and the operations that may be inferred by the context of the story. These type of “numberless word problems” are being used more and more by practitioners in order to deepen children’s understanding of the concepts involved.

Another key step in the RUCSAC approach is the ability to create to show what you know, where the child makes a representation of the word problem in another form. Bar models are ideal for use with operational word problems. Introduced initially in Operation Maths 3, the use of bar models is developed through Operation Maths 3-6 to include bar models suited to other types of word problems.

Empty number lines can also be used to represent addition and subtraction problems (see below). In the senior books, the children will use both strategies to represent word problems and compare and contrast the two strategies. Ultimately, it is hoped that the children will use the strategy that they are most comfortable with. For more information on problem-solving strategies please consult the guide to problem-solving strategies across the scheme in the introduction to your Teachers Resource Book (TRB) or read on here.

Communicating and expressing thinking

Being able to explain your mathematical thinking is a very powerful tool, and one that can greatly aid the learning and understanding of both the speaker and the listener(s). Encourage the children to verbalise how they did their calculations (mental or written) to provide you with a window on their thinking. When talking about decimal numbers, encourage children to use fractional language as opposed to decimal language, i.e. ‘6 hundredths plus 4 hundredths is ten hundredths’ etc.

Another way to communicate and express thinking is via jottings. These are informal diagrams that both show and support thinking, and when used as a part-mental approach, serve as an intermediate stage between concrete materials and the abstract calculation. Their use should be encouraged as much as possible (e.g. “use jottings to show me your thinking”) until the child is confident enough to do the whole calculation mentally or using a traditional written form. The main jottings used in Operation Maths are empty number lines (pictured above) and branching (pictured below) to show part–whole relationships and/or explore compensation.

Online Resources

  • Operation Maths Digital Resources: As always don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie. Tip: look at the footer on the first page of each chapter in the pupil’s book to get a synopsis of what digital resources are available/suggested to use with that particular chapter.
  • Estimating sums and differences
  • Hit the button: Can used to play various games involving doubles and number bonds of whole and decimal numbers
  • Thinking Blocks: an interactive resource that enables you to build bar models to solve problems. This is a great way to practice the different types of bar models for addition and subtraction when you are unfamiliar with this visual strategy.
  • That Quiz – Arithmetic: Use this quiz to practice different types of addition and subtraction calculations. To find out more about the potential of That Quiz across all strands and subjects, please read on here.

Further Reading:


Number Talks & Operation Maths

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

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

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

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

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

What does a Number Talk look like?

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

Number Talks Approach

Operation Maths & Number Talks

1. The teacher presents a number sentence to the class; the students are given thinking time to mentally solve it. The horizontal number sentences in Operation Maths can in themselves inspire or be used as the basis for a number talk. For example, similar number sentences to the ones shown below this table were used to encourage the children to use compensation to solve calculations.
2. The students start with one fist to their chest;  they make a “thumbs-up” on their chest to show that they have found an answer. They then use the remaining time to try to think of another way/strategy which they then indicate by putting up a thumb and a finger, and so on. While I initially used this “fist and thumbs-up” system when collecting answers, after multiple times hearing “I had the same answer as Jack/Jill”, I returned to my preferred tool of using the Operation Maths mini-whiteboards, (to maximise on participation and honesty regarding answers).  It is important to insist that the MWBs are not to be used at all for working out, all of which is to happen in the heads, rather they should only be used to record the answer(s).
3. The teacher asks a number of children to volunteer their answers and all given answers are recorded on the board.
4. The teacher asks a child to “defend their answer”/”explain their strategy”. For the children to explain clearly, they need to have the correct mathematical language so that all listeners can follow their thinking. Thus, children who have been using the Operation Maths programme are typically better able to express their thinking using the correct mathematical language and terminology that is being emphasised throughout these books.
5. All strategies are recorded on board by teacher, using visuals where possible to make the strategy less abstract for the other listeners. Many of the visual strategies that are specifically recommended to be used are ones that already used extensively throughout Operation Maths eg frames, empty number lines, bar models (referred to as part/whole models), arrays and  area models. Branching is another visual way to demonstrate strategies particularly when partitioning (breaking into place value parts) /or compensation is involved.
6. The children agree on the “real” answer. Depending on the range of possible answers given, the children can also be asked to identify any unreasonable answer from those suggested and explain why they think so. This in turn encourages them to apply the variety of estimation strategies taught in the Operation Maths programme

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:

 


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

What is a ten-frame?

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

They can help children:

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

 

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

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

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

 

Horizontal or vertical?

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

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

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

 

Four relationships for number sense

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

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

 

 Spatial relationships and subitising

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

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

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

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

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

 

One and two more than/less than relationships

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

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

 

Benchmarks of 5 and 10

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

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

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

 

Part-part-whole relationships

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

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

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

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!


Enabling Computational Fluency

 

‘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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