Rounding Defamiliarisation

In literature, defamiliarisation is where the typical is made unfamiliar to nudge the readers’ noticing and subvert their perceptions to consider different perspectives.

Maths teachers do this all the time. Variation theory, minimally different questions, non-examples etc. are all concerned with singling out precise aspects from the myriad of conditions in a given question or content area.

This post outlines how I’ve used defamiliarisation in my teaching of rounding this half term.

Firstly, Sudeep at Boss Maths has a Geogebra applet in their rounding and truncating resources. (It’s great and it’s here)

We’ve been using this to do lots of low-stakes questions with some backwards fading for groups that have been struggling.

Using the slider at the side, I can partially complete questions and have pupils finish the rest on mini whiteboards
This proved to be the most challenging stage for a significant amount of kids.
We’ve been using different language here, instead of consecutive multiples, we’ve been asking for the section of a number line starting at 0 in intervals of 100 – which sounds clunky at the moment, but is based on a lot of work we’ve been doing around number lines.

In using this, combined with examples and practicing minimally different questions, pupils have become familiar in seeing rounding as a discussion of closeness.

Rather than crack right on with significant figures, we’ve spent a bit more time just consolidating.

How much are we REALLY thinking about closeness if we’re only looking at rounding to decimal places and powers of ten?

Using non-standard rounding parameters hones in on the concept of closeness, in much the same way we used different bases to make implicit place value understanding explicit.

To start with, the task bellow asked pupils to finding the midpoints of the circled numbers on each number line.

This is to over-learn finding midpoints for what we’ll need them for later (between multiples for rounding), but has merit in and of itself (the amount of pupils that get to KS4 (and even KS5) without being able to confidently find the midpoint of numbers can cause stumbling blocks in coordinate geometry and statistics).

Once we’d completed these questions, then we began to discuss them. What did we notice? What did we think about the links between the questions? How could we write questions that used the same structural properties? The picture below shows prompts for some of the connections in the task.

We followed this up with a discussion about different aspects of different number lines and the relationships between intervals and scales and differences and lengths etc.

After a few worked examples with sketching number lines, most pupils really struggled with the idea that 4 to the nearest 8 would be 8. Even though we’d discussed why 5 to the nearest 10 is 10 when we first began looking at rounding as closeness, this aspect of defamiliarisation made students acutely aware of proper actual midpoints and that rounding up is a convention.

Pupils then tackled the task below, with the question prompts in the images that follow.

Defamiliarisation in mathematics teaching is our bread and butter, and tweaking questions to nudge thinking works best if we explicitly tell pupils we’re doing this and why.

It’s difficult to design tasks that do this that don’t rest on just patterns of answers and change the focus from the structure of the maths to a more surface level noticing. Having prompts once we’ve gone through answers and modelling what it looks like to be a proper maths nerd helps pupils build those habits.

All resources used, and solutions, are linked in the slides at the top of the post, and a massive thank you to Sudeep at Boss Maths for creating such an incredible digital manipulative.

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