How can we help students to build conceptual understanding? This is a puzzle I'm constantly thinking about. There can be exceptions or nuances to concepts that aren't always obvious at first, e.g. 5 - -3 is equivalent to 5+3, but the negatives in -5-3 don't get turned into '+'.
John Mason and Anne Watson have written extensively about mathematical thinking and pedagogy in relation to building conceptual understanding. Here they explain:
"there are ‘edges’ to mathematical ideas which are worth exploring."
How might we get students to explore these ‘edges’? Giving examples or techniques, will often not be enough. Students might accept the ideas they are presented with, but this is not the same as understanding. Instead, students need opportunities to construct and reconstruct examples and to reflect on the edges of those examples, i.e. what can be counted as a true example or not. What's The Split? is a new routine I’ve been working on. It invites students to think critically about examples they are presented with and to justify their ideas.
Here's one approach to running What's The Split? as a whole class activity:
Present 4 items to the class and ask: how might you split these into 2 groups?
Give students individual time to think before sharing and justifying responses
Use the 'Think about' questions to encourage students to go further and clarify their understanding.
Depending on your context, you could also run the routine:
in pairs or individually
as a diagnostic tool or quick formative assessment
as short, regular tasks anywhere between 5 min and 20 min
If you're familiar with the routine, 'Which One Doesn't Belong?' you may notice some similarities. Here's Matt Skoss (Project Manager, Mathematical Association of NSW and Possum Educational Services) on how the two compare:
"In WODB, the challenge is to try to find an attribute (missing or present) that's unique to an individual item. In What's The Split, the idea of the 'moving’ split brings in a fluidity to thinking that is also quite challenging, requiring in-the-moment adjustment to someone’s thinking. It also allows less confident learners to offer their thinking, but could ramp up to quite sophisticated thinking. Someone might notice that two of the numbers are one less than a cube number, for example."
For more examples of What's The Split? covering different parts of the curriculum, check out this google drive folder. Feel free to also add your own for others to use. I'd love to hear what you find works or doesn't work with your students, so we can make this routine work better for all.