There’s more to subitising than meets the eye!

Continuing with our ongoing theme of sharing the guidance documents that we are producing to support the new Highland Numeracy Progression Summary Documents, today will look in more detail at the document on subitising.

We frequently see the perception that subitising is just about recognising how many dots are on a five frame or tens frame or perhaps in an irregular pattern with the goal being just to know how many dots there are.  There is so much more to subitising than this and through rich discussion important mathematical ideas including can be developed including but not limited to: spatial awareness, estimation, patterns and relationships, mathematical properties and relational thinking.

The guidance document explores subitising in more depth including distinguishing between two types of subitising:

  • Perceptual Subitising 
  • Conceptual Subitising

As well as subitising in different ways including: spatial, temporal and kinaesthetic subitising.

We also explore ways in which conceptual subitising can be used to support a range of different ideas including:

  • Composition and Decomposition
  • Addition and Subtraction
  • Basic Facts
  • Place Value
  • Multiplication and Division

We discuss how subitising can be made more or less complex depending on the types of arrangements used.

We also discuss common misconceptions and things to look out for.

And finally, there is further exemplification of some of the ideas that subitising builds the foundations for.

Regardless of whether subitising is something new to you or you are very knowledgable about it already, we hope there is something in the document to start you on your journey or an idea that you hadn’t considered before that gets you thinking.

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Maths Week Scotland – One month to go!

28th September – 4th October – Pin your school on the map.


Highland schools are beginning to pin their school on the Maths Week Scotland website .   This week is an opportunity to celebrate the importance, enjoyment and contribution that maths makes to our lives.  Schools plan from the big scale – with a week of spectacular activities and maths parties, to the small scale where they plan a day of maths fun, or a special lesson.  Whatever you choose to do during the week – let’s celebrate together and remember to pin your school on the map!

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Mathematical Properties – What underpins children’s informal strategies?

This post, which follows on from the last two, forms a series of posts that aims to highlight some of the guidance documents that we’ve included in the new HNP Summary Documents for each year.  If you haven’t read the last two, you can find them here:

Making sense of part-whole strategies for addition and subtraction

Progression in Number Ranges at First Level (First** and First***)

Today’s document, Properties of Addition and Subtraction, has several aims:

  • To raise awareness of the mathematical properties that underpin the informal strategies that pupils use.
  • To highlight some of the main features of the guidance document.
  • To try and bridge a gap between Primary and Secondary.

The document is quite lengthy and it’s not intended that you read it straight through from start to end but use it more as a guide or reference document to support your planning, teaching and understanding of mathematics.

I’ll be honest, when I started teaching, my knowledge of the properties referred to in the document, some of which include:

  • the commutative property,
  • the associative property,
  • the distributive property,

was limited.  I may have had an implicit awareness of them from school, although my schooling was very much of the rules and procedures style and asking why something worked was met with the response that I didn’t need to know why, I just needed to learn it.  I may have covered them very briefly on my teacher training course but again, I’m not particularly aware of them having much prominence there although they are clearly in the main texts that were part of the course: Haylock was one and Frobisher was the author of the other.  I believe these are still the main texts used on courses today… although they may be several editions on!

So, if you read the document and think that some of the content is a bit heavy going… fear not!  I would have also felt that way at one point too.  That being said, I do feel it is fundamentally important that teachers at both primary and secondary have a deep understanding of the properties and how/where they are used.

From experience, although this is purely speculation, a lot of the recurring themes where children get stuck or have misconceptions appear to stem back to a lack of understanding about these mathematical properties.  I believe a lot of this could be mitigated if both pupils and teachers being more aware of the properties.  Algebra would be a good example of this but another good one is pupils frequently trying to solve problems like 52 – 25 but getting an answer of 33.  The document highlights issues like this and connects them back to the properties.Screenshot 2020-08-28 at 09.41.29.png

An additional point to make here is that from a really young age, an implicit awareness of the properties can be fostered as part of play based learning.  Ideas for activities that support a developing awareness of the properties is included within the document – this includes ideas for pupils in Nursery settings but extends to activities for older pupils too.Screenshot 2020-08-28 at 09.42.00

The activities don’t need to be dry, and if teachers really understand the properties, incorporating understanding of them while pupils are exploring different strategies can promote curiosity as to how on earth it’s possible to solve a problem in that way or how someone has solved a seemingly complex problem in a matter of seconds!  Ideas are given about how to support this kind of discussion and how to extend pupils’ understanding of these properties through the development of relational thinking (among other things).Screenshot 2020-08-28 at 09.43.04.png

And there is, hopefully, a clear explanation of what each property is and how it links to the informal strategies that pupils use.  This ties back well do the guidance document on part-whole strategies for addition and subtraction.Screenshot 2020-08-28 at 09.42.42

The final point I want to make is about bridging the gap between primary and secondary.  My feeling is, and this is a generalisation based on my own experiences and certainly not true in every case, that secondary teachers have a good awareness of the mathematical properties but not the informal strategies that pupils use and primary teachers have a better understanding of the informal strategies that pupils use but lack the understanding of the mathematical properties.  If this gap could be bridged, by both parties, then I think primary teachers would be in a better position to help guide the use of pupils’ informal strategies to increasing efficiency and sophistication, through greater awareness of the mathematical properties at play.  If secondary teachers were more aware of the informal strategies that pupils use and the mathematical properties that underpin them, it would become more apparent how these properties are the same properties at play when pupils learn algebra and so these ideas, that pupils often find so hard to grasp, can hopefully be attached to something that pupils are more familiar with and can generalise from.

I’d be interested to know people’s thoughts on this and whether this reflects other people’s experiences or not.

As ever, if anyone spots any mistakes or anything that is not accurate within the documents please do let me know so I can address this.

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Progression in Number Ranges at First Level

The post yesterday was related to making sense of part-whole addition and subtraction strategies that pupils use and showed some examples of these with concrete materials and visual representations as well as exploring the mathematical properties that underpin them.

The guidance document I’m going to link to today has a progression of number ranges at First** and First***.  The two documents could be used alongside one another to support your planning.  (We will hopefully create one of these for Second Level too, in due course.)

An overview of the document is provided below including some important points to keep in mind.

  • The numbers are all presented as bare numbers e.g. 37 + 8 (rather than in word problems or other contexts) this is purely to illustrate the numbers involved and not a suggestion that you should only provide examples in this way.
  • These numbers should be used as part of a variety of problem types as has been exemplified elsewhere, these include problems that involve comparison (and looking at the difference), situations where you are breaking a whole into parts (part-part-whole) and problems that involve the more common problem types of increasing or decreasing a quantity including varying which part of the problem is unknown.
    This also includes varying the type of practice task you give pupils.  Presenting tasks in different ways will help draw out different relationships or underlying properties.

    Screenshot 2020-08-21 at 09.40.14.png

    Screenshot 2020-08-21 at 09.40.26

  • When you use the number ranges, they should include a variety of equation representations e.g. 15 = 5 + ? or 6 + ? = 5 + 10 as well as the standard representation 5 + ? = 15.  This helps pupils make better sense of the actual meaning of the equal sign.  This should also extend to the use of inequality symbols e.g 5 + 10 < 6 + ?

Screenshot 2020-08-21 at 10.07.17

  • At each level there are several main milestones, for want of a better word, in terms of solving a particular type of problem.  These have been broken down into smaller steps but this is not necessarily so that teachers teach in these micro-steps one step at a time.  It’s more to help teachers understand the smaller skills involved in achieving the end goal.  They can then use this to help them make sense of how they are going to structure their teaching, what type of practice type might be useful, what connections they will make and how they will support pupils to understand the mathematical properties.Screenshot 2020-08-21 at 09.12.38Screenshot 2020-08-21 at 09.13.48
  • And last but not least, there is a summary of the types of numbers that lend themselves to the different strategies that were explored in the guidance document that was explored yesterday.Screenshot 2020-08-21 at 09.14.12.png

From my experience, a lot of textbooks don’t promote this level of thinking and the way the questions are organised and the types of representations they offer often limit the opportunities for really understanding and exploring the mathematics in an enjoyable way that promotes pupil questioning and curiosity.

Hopefully this document alongside the other ones we are sharing will mean people feel better equipped to understand the progression of learning and also plan meaningful, engaging experiences for their pupils.

As always, if anyone spots any mistakes or disagrees with anything then please do let us know!

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Making sense of part-whole strategies for addition and subtraction

Yesterday we shared the (draft) materials for the HNP Summary Documents for the rest of First Level.

Over the next few days, we’ll highlight a few of the guidance materials and example activities that are contained within the Summary Documents to help people know what is available and how it can help.

Today we’ll look at the the document:

Addition and Subtraction: Part Whole Strategies Summary Guidance First**/First***

Firstly, it’s important to know that if you are teaching pupils below or above these levels the document is still relevant.

  • For those working with pupils below First**, it is important to know what’s coming next so you can make sense of why certain skills that your pupils are working on are relevant and how they feed into future strategies.
  • For those working with pupils above First***, the strategies that pupils will use to solve addition and subtraction problems are the same, so this document will still help you to make sense of content relevant to Second Level and above.  The difference between First Level and Second Level and above will be related to:
    • The number range the pupils work within.
    • The pupils’ ability to more effectively choose an efficient strategy based on the numbers within a particular problem.  It should be noted though, that there is still an expectation at First Level that they are reflecting on and justifying their choices.
    • The pupils’ greater understanding of the mathematical properties that underpin these strategies and their subsequent application to algebra.

Within the document you’ll find information related to the following areas.

General Guidance which includes:

  • A general look at how to approach pupil understanding of part-whole strategies (although ideas for teaching is covered in more detail in another document).
  • Information about whether pupils need to learn all the strategies!
  • Information about language and making connections.

Jump and Split Strategies

  • Have you ever wondered why your pupils answer problems like this: 63 – 25 with incorrect answers like 42?
    This is explored in the Jump and Split strategies section.

The Part-Whole Strategies:

The part-whole strategies, that pupils might use to solve addition and subtraction problems, are explored in more detail including:

  • Prior knowledge that may be required.Screenshot 2020-08-20 at 15.11.39
  • Common misconceptions.
  • Representations with concrete materials and visual representations that might support understanding.Screenshot 2020-08-20 at 15.13.54.png
  • The mathematical properties that underpin certain strategies.
  • A video example of a pupil using the strategy to solve a problem mentally.
    While most of the videos show examples of number ranges and strategies at First** and First***, this particular video is highlighting an example of a pupil using a strategy to solve a problem in a number range that is more closely related to Second***.


We hope you find the documents useful and we’ll share some more detail about what’s inside some of the other materials over the next week or two.

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Highland Numeracy Progression Summary Documents: First Level

The Highland Numeracy Progression Summary Documents for the rest of First Level are now available on the Shared Drive, as well as on the blog.  They will be added shortly to the Highland Digital Hub.

A huge amount of work has gone into the documents and the additional supporting Guidance Documents.  We hope that people find them useful for their own knowledge and understanding as well as to support their teaching.  We’ll share some of the features of the Guidance Documents over the week to help people navigate their way around some of the additional materials.

As a reminder, if you haven’t already, please read the HNP Summary Documents: Overview and Rationale before using the documents so you understand them better and use them appropriately alongside other resources we have available.

These are ‘live’ documents which means additional supporting documentation and resources will be added on an ongoing basis.  As such, we suggest you use them online (or regularly check that you have the most up to date version) rather than download them.

Although we are adding the rest of First Level, the section on Multiplication and Division is still being constructed and will be added to the document soon.

We are currently working on Second Level and hope to have this out within the next three weeks.


First** HNP Summary Document.png


First*** HNP Summary Document

As always, if anyone happens to spot any mistakes, please do let us know and if you disagree with any of the points we’ve made then we’d be more than happy to have a discussion about this.  

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Maths Week Scotland Small Grants Fund

Maths Week Scotland: 28th September – 4th October

Maths Week Scotland is an opportunity to celebrate  the importance of maths in our everyday lives.  Applications are now open for the Small Grants Fund.  Schools can apply for a small amount of money to go towards celebrating this week.  Funds can help with:

  • Outdoor learning
  • Digital resources
  • School Events
  • Addressing barriers to innovation
  • Recognition / awards for taking part

The closing date for applications is the 26th August.

For more information click the link Maths Week Scotland Small Grants Fund.


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Socially Distanced Diagnostic Assessments

I hope everyone has had a restful summer and is looking forward to getting back to school and the kids.

Whenever we are looking to teach, it is useful to get a gauge of what our pupils can already do and where they have gaps, misconceptions or content they may not have been taught yet (or forgotten).  To help with this I’ve amended the format for the diagnostic assessments so these can be delivered at a ‘social distance’ without the need for the problem cards and other materials.  With such a gap between when pupils were last ‘formally’ in school, it may be more useful than ever to get an accurate gauge of where they are to help them move forward.

All the content is on the new Shared Drive which you can request access to here.  Additional content will be added over time.  Please use your Highland Google login details when requesting access (e.g.

If you haven’t already accessed the diagnostic assessment online training, we strongly suggest you do this before using the assessments so you have a better understanding about the rationale behind them, how to implement them and how to interpret the information.  A guide to this training can be found here.

Direct links to the diagnostic assessments that have been put into a socially distanced format can be found here:

The multiplication and division assessment and the fractions assessment have not been included as we are in the process of updating them.


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Summer Reading Ideas

I have just finished writing a long list of book suggestions for someone on Facebook so I thought I might as well post it on here in case anyone else is looking for some teaching related summer reading ideas!

The list comes from quite a wide range of places and includes different approaches to teaching maths and some that are linked to commercial products.  I’m not advocating any single approach or commercial product but do feel that reading widely and comparing different viewpoints and research is really useful to get the full picture.

I’ve written a very brief blurb about each to give you a bit more information.  There are also a couple that aren’t directly related to maths but are still very relevant.

In no particular order, here they are:

Mathematics in the Early Years:

Big Ideas of Early Mathematics (Erikson Institute)

  • It’s easy to read with a mix of practical ideas and thoery.
  • This has some really nice ideas for the Early Years with content also relevant for P2.  It gives a good sense of some of the important foundations for maths (that are perhaps sometimes overlooked or rushed) and may help people understand why some struggle later on and how enriching Early Years maths education may help prevent future maths difficulties.
  • It is written by the people behind the Erikson Institute Early Math Collaborative which is a brilliant website.

Learning and Teaching Early Math: The Learning Trajectories Approach (Clements and Sarama)

  • It’s easy to read.  The book is structured so the first section in each chapter is theory based.  This is followed by a table which has a progression in learning from 1-2 years old to 8 years old (this varies a little depending on the area).  The progression also has some activity suggestions.
  • It is written by the people behind the Learning Trajectories website.
  • It’s quite a lot more expensive than some of the other books on this list (but is very useful).

Young Children’s Mathematics: Cognitively Guided Instruction in Early Childhood Education

  • It’s easy to read with a mix of practical ideas and theory.
  • It is linked to a series of other books all related to Cognitively Guided Instruction (see below) but this one is specifically related to the Early Years.
  • If you’re not already on it, there is a Cognitively Guided Instruction In Scotland Facebook page as well.  This was set up by Dr. Lio Moscardini.

Teaching and Learning Early Number (Edited by Ian Thompson)

  • It’s easy to read but it’s probably more theory based than some of the others mentioned above.  It does have practical applications though.
  • It’s good for getting you to think about the ideas that are important for teaching maths.
  • It’s a little different from the others as each chapter is written by a different author.

Teaching Mathematics 3 – 5 (Sue Gifford)

  • It’s easy to read but it’s probably more theory based than some of the others mentioned above.  It does have practical applications though.
  • It’s good for getting you to think about the ideas that are important for teaching maths.
  • Sue Gifford writes a lot of the articles on the NRICH website related to Early Years Education.

General Primary Maths:

Children’s Mathematics: Cognitively Guided Instruction

  • Also part of the Cognitively Guided Instruction series (CGI) series of books.
  • It is good if you want a general overview of CGI at a range of levels.
  • It’s easy to read with lots of practical ideas for application.

Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School

  • Also by some of the authors of the other CGI books.
  • Excellent if you want you, as a practitioner, want to understand more about the properties of behind some of the operations and therefore support your pupil’s to better understand this.
  • It will support you to get your pupils thinking more mathematically and go far beyond just answering pages of sums and having these marked as either right or wrong.
  • It’s probably a bit heavier going than some of the others but still easy to read (in my opinion) and very worthwhile!

Choral Counting and Counting Collections: Transforming the PreK – 5 Maths Classroom

  • Also from the authors behind other CGI books.
  • It’s great if you want to get a bit more detail about how to introduce Choral Counting and Counting Collections into your classroom.
  • It’s easy to read with lots of practical applications.

Books related to fractions:

Developing Fractions Knowledge (Hackenberg, Norton and Wright)

  • This is quite heavy going but it is very interesting and very useful and will probably give you a lot more in depth knowledge about how to teach fractions effectively.
  • It’s written by some of the same people behind other Maths Recovery books (but in my opinion is not quite as easy to read as the others).
  • As well as the theory side of things, it also includes questions for assessment purposes and activities – these could be easily implemented into your classroom.

Extending Children’s Mathematics: Fractions and Decimals – Innovations in Cognitively Guided Instruction

  • As the title suggests, this is another in the CGI series and is written by some of the same authors as the other books but specifically related to fractions and decimals.
  • It’s easy to read and has lots of practical applications as well as understanding the progression in learning.
  • It complements the Maths Recovery book above well.

Books related to Mastery:

Mastery, although it’s been around for a very long time seems to have made a bit of a resurgence recently.  From some of the things that I’ve seen/read on the internet, a lot of people appear to be reading the headlines and taking quite a superficial look at it.  Here are a few books if you want to find out more.

Teaching for Mastery (Mark McCourt)

  • It’s easy to read and gives an overview of the background behind mastery.
  • There is content related to both primary and secondary.
  • Mark McCourt is behind LaSalle Education and Complete Maths.

Mastery Learning: Theory and Practice (Edited by James Block)

Implementing Mastery Learning (Thomas Guskey)

Although I’ve read work by both authors/editors above, I haven’t read these books but someone has kindly let me borrow them.  I’ll update this with more information when I’ve read them.

Books related to Mindset:

I feel that people often take the headlines of some of the information shared on Mindset and as a result it’s often implemented in quite a superficial way.  (In a similar way to the comments I had about Mastery.)

Mindset (Carol Dweck):

  • Not specifically related to maths but a very interesting read about ‘mindset’ in general.

The Elephant in the Classroom and Mathematical Mindsets (Jo Boaler):

  • Both explore ‘mindset’ specifically from a maths perspective.
  • They are both easy to read and provide an interesting read.
  • Both, but more so the Mathematical Mindsets book has practical ideas that you could use in the classroom.  There are a mix of activities for both primary and secondary.
  • Jo Boaler is behind the YouCubed website.


The Number Devil:

  • This is actually a children’s book.  It would be good for upper primary or the first few years of secondary.
  • It is about a boy who hates maths and then meets ‘The Number Devil’ in his dreams who transforms the boys perceptions of mathematics.  It explores quite a lot of common areas in mathematics in an engaging way.
  • It would be a good read as a class novel and to explore some of the ideas along the way.
  • As an adult, I enjoyed reading it.

Closing the Vocabulary Gap (Alex Quigley):

  • This is not specifically related to maths but definitely has connections and is very relevant from a Closing the Poverty Related Attainment Gap point of view.
  • It has lots of practical applications across a range of subjects.
  • It would be suitable for both primary and secondary.
  • Alex Quigley did a podcast on the Mr Barton Maths Podcast if you wanted to listen to that instead or before deciding whether to read the book.
  • The book in my opinion provides a lot more detail than the podcast though but the podcast was a great starting point.

Make It Stick (Brown, Roediger, McDaniel):

  • This book is basically about memory and how we learn.
  • It’s easy to read and is absolutely fascinating (in my opinion).  I couldn’t put it down.
  • There are huge implications in terms of how a lot of teaching is usually carried out… and how it could be organised/structured much more effectively.
  • This has implications for your own learning (as an adult) as well as how we teach our children.

I could go on, there are many more wonderful books that I could mention but that list is probably long enough as it is.  I’ll maybe write another list later in the year as an update!

They are all great in my opinion, it’s quite hard to choose where I’d start as they are all quite different.

If you do read any, then leave a comment below to share what you thought of it.
If there are any that I haven’t included but you think should have been on the list, I’d love to hear your thoughts.

Happy reading whatever you decide to go with!

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Highland Numeracy Progression Summary Documents

The information outlined below is now contained within its own page.  The page HNP Summary Documents is organised as a sub-page of the Highland Numeracy Progression main page (see image below).  All updates to these documents will appear there.

HNP Summary Documents Blog Page.png

The following materials (see images below) have been put together as a summary of learning for each stage.  These should be used alongside the main Highland Numeracy Progression document.

We encourage all practitioners to read the HNP Summary Documents: Overview and Rationale before using the documents so you understand these documents better and use them appropriately alongside other resources we have available.

These are ‘live’ documents which means additional supporting documentation and resources will be added on an ongoing basis.  As such, we suggest you use them online (or regularly check that you have the most up to date version) rather than download them.

The documents below include Early*** and First*.  We will add the documents for the other stages as and when they have been finalised.  Click on each image below to be taken to a PDF of the document.

HNP Summary Document: Overview and Rationale

HNP Summary Documents Overview and Rationale






HNP Summary Documents: Early Level

Early*** HNP Summary Document   

HNP Summary Documents: First Level

 First* HNP Summary Document

These documents are also available on the Highland Digital Hub:Highland Digital Hub

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