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.
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, 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 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.
This has always been one of those problems that I have loved exploring. I recently found a website that will model the outcome in a brilliantly beautiful way. Here’s a little more about it from the Birthday Problem Wikipedia page:
In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday.
By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.
It may seem surprising that the probability is above 50% when there is a pair with the same birthday for a group as small as 23 individuals. This is made more plausible when considering that a comparison will actually be made between every possible pair of people rather than fixing one individual and comparing him or her solely to the rest of the group.
The Pudding’s Russell Goldenberg has created this website to demonstrate this paradox.
Have a visit, have a play and contribute to the data.
I would absolutely use this as a hook the next time I need to tackle statistics with an upper Key Stage 2 class!
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.
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!
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.
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.
Feel free to share with colleagues – I would love to know how they are being used.