Start as you mean to go on!

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


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:

 


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


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


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