Nonexamples: Conceptual Variation

A great way to think about conceptual variation is to ask children when something doesn’t fit into a specific set of criteria – knowing not only the properties of a maths topic but being able to explain why particular objects don’t belong in that topic too is a feature of maths mastery.

Conceptual variation means the opportunity to work on different representations of the same mathematical idea. This might be, for instance, looking at multiple representations of a number with Diennes equipment, place value counters, Gattegno grids, place value grids, arrow cards etc. These numerous representations demonstrate to pupils the different conceptual ideas that underpin a mathematical concept.

So, in the context of place value, some will reveal the quantity or value of a digit, some will show the importance of the position of a digit, others will support the order of the number and some will reveal the additive or multiplicative nature of place value.

The use of conceptual variation can be applied to all areas of the mathematical curriculum.

One website I’ve found to aid in this aspect of conceptual variation is

The site has a range of mathematical topics to choose from and will display a variety of appropriate images of both examples and non-examples relating to it. There are two modes currently: question and compare. – Question mode

I feel the initial question mode would be useful for opening lessons, possibly mid-way through a unit of work. They would also be a wonderful resource to develop and assess children’s reasoning skills. – Compare mode

Compare mode, however, is something I would use to introduce a new concept – perhaps even just displaying examples and non-examples side by side as above and asking children what they notice.


Numberblocks: NCETM Support Materials

Numberblocks is a pre-school BBC television series aimed at introducing children to early number.

These NCETM materials use each episode as a launch pad. They are designed to assist Early Years (and also Year 1) practitioners to move on from an episode, helping children to bring the numbers and ideas to life in the world around them.

You don’t even need to register with the NCETM to use them!

The materials are designed to be used in conjunction with the Numberblocks episodes. They highlight and develop the key mathematical ideas that are embedded in the programmes. Each set of materials comes in the form of a PowerPoint file, and includes the following features:

  • The episode description summarises the story and the key things that happen
  • The maths in the episode explains the key mathematical concepts that are featured in the episode
  • Using mathematical language – because it is important that practitioners model precise and correct mathematical language, there are suggestions of key sentences that you might use and have repeated; they provide a language structure to connect each mathematical idea to different contexts. Children will initially use their own language to talk about the mathematics, and will develop correct and precise language if this is modelled by adults.

Overviews of each series, their storylines and the mathematics addressed, are available as downloadable PDFs from the link below.

More materials are available on the CBeebies’ Numberblocks page, also linked below.

Link: NCETM Numberblocks Materials

Link: CBeebies’ Numberblock Page

Simplifying addition/carrying with regrouping

I’m not quite sure why I’ve never thought of this before now, but… using sticky notes to highlight the partitioning of a number after an addition to show how to deal with numbers larger than 9 is a complete revelation to me!

This idea (and associated images below) is from The Classroom Key.

Show students how to add up the numbers in the ones column and write the answer on a sticky note.  In the beginning, it is probably worth labelling the left side of the note “tens” and the right side “ones.”

Use a pair of scissors to cut the note in half vertically.

Put the “tens” half of the note on top of the tens column.  Put the “ones” half underneath the ones column.  Add up all the numbers in the tens column and there’s the answer!

I will certainly be using this with my pupils in future when introducing this concept. I’d love to hear how you get on if you try it too.

Link: The Classroom Key

Free Download: Mathematical Vocabulary Flashcards

The following ten files are a straightforward set of mathematical vocabulary flashcards. I have grouped them by topic and have written on in small lettering where in the curriculum the term is introduced.

A sample page from the addition and subtraction group is shown below. All of the pages share the same design.

Page 1 of “Addition and subtraction vocabulary.pdf”

Feel free to share with colleagues – I would love to know how they are being used.