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 nonexamples.com

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.

nonexamples.com – 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.

nonexamples.com – 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.

Link: nonexamples.com

Opportunities to notice – developing number sense and reasoning

I have long been a fan of Steve Wyborney’s blog where he shares plenty of engaging and different ideas to develop mathematical thinking. Although based in Oregon, the ideas are still very relevant to the UK curriculum.

I recently came across his use of dot patterns to promote ‘noticing’ – not something we necessarily spend a lot of time on, but I now wonder why we don’t.

One big focus in mathematics teaching currently is the use of small step and space to let children think. This sort of activity undoubtedly lends itself towards these ideas.

Steve has created 10 pages of dotted patterns, the purpose of which is to promote a wide variety of partitioning, seeking different perspectives and recording observations in ways that can be very useful. I would guarantee there will be children in every class who see a different pattern or way of breaking things up to draw out different mathematics every time.

He recommends using 18 copies of the same pattern with the space underneath being vital to allow plenty of working room and many opportunities to explore the same thing in different ways.

The following image is from his explanation video of how to use the patterns (found here), demonstrating 4 ways to investigate one such pattern.

For a greater challenge, once children are used to working with these, you could develop more complex images with different coloured dots – each colour being worth a separate value – to explore larger numbers, perhaps. Each dot doesn’t need to necessarily only be worth 1, either.

The 10 patterns and some further information can be found at the link below.

Link: 180 Opportunities to Notice