Blog: John Jackson, Secondary Development Lead
Where’s the challenge? vs. what’s next? ‘Depth over breadth’ in practice
Your Year 7 class are flying through their work on calculating missing angles. They’re finding missing angles around a point, on a straight line and in triangles. Vertically opposite angles? Isosceles triangles? No problem.
They’re even giving rich answers like “…angle ABC must be 32o because the other two angles add up to 148o and the angle sum of a triangle is 180o” (as opposed to just saying “it all adds up to 180”).
So what next?
Parallel lines? Angles in polygons?
Here at Mathematics Mastery, we don’t think so. We believe students should be challenged by developing their depth of understanding of a topic – instead of being accelerated onto (superficially) harder content.
In this instance, this might include investigating measuring angles in gradians. Maybe radians won’t then be such a shock at A Level. Or it might even inspire a career as a chartered surveyor or an engineer in the British military.
We could also tackle problems like this:
In an isosceles triangle, one angle is four times the size of one of the others. Calculate the sizes of the angles in both possible triangles and construct both triangles accurately, choosing your own side lengths.
We could also ask our students to generalise here – if instead of 47o and 36o, the angles are x and y, calculate the size of the missing angle.
But why do this instead of giving students’ a head-start on new curriculum content?
As we’ve mentioned before, developing a depth of understanding supports students in building their problem solving skills, improves flexibility and fluency, and helps make links between topics.
This helps students retain their understanding, which is hugely beneficial when they reach GCSE, A Level – and beyond!
As mathematics teachers, many of us have witnessed students hitting a GCSE ‘brick wall’, which they struggle to break through when studying A Level mathematics. Even for the very high achieving students, a lack of deep mathematical understanding can suddenly be a huge inhibitor when studying at A Level standard.
However, we understand that developing depth tasks is great in theory but requires a learning curve – and persistence – in practice.
So how we can get better at generating ideas for depth challenges?
When I design Mathematics Mastery depth tasks, I start by looking at our lesson plans and then ask myself a few questions.
- Could students make generalisations about areas of this topic?
- Is there more than one way to solve problems like this?
- Can I create puzzles involving this topic?
- Are there any strange or unusual examples e.g. self-intersecting quadrilaterals?
- Can I combine the new topic with other ideas we’ve already covered?
- Are there other perspectives students could consider e.g. different bases or systems of measurement?
Now, we’re not pretending this is an easy switch to make. I certainly spent a significant amount of time in the classroom stretching students through acceleration. But too often this led to students either being unable to solve problems or failing to retain what they’d learnt.
We have to shift our mindsets and consider ‘what’s interesting or challenging about this?’ rather than simply ‘what’s next?’
One of the benefits of this approach is, it generally doesn’t require you to create loads of questions or separate worksheets. Instead, you just adapt a problem or think of an additional, meatier, question. This means depth tasks can easily become part of lesson planning, particularly within team planning time and ultimately saving you time.
Here’s a standard problem about finding the mean.
Find the mean of 3, 5, 5, 6, 7 and 10.
Here are some ways you might add challenge through depth.
- Change two numbers so that the mean stays the same.
- Change all the numbers so that the mean stays the same. Can you find three different ways to do this?
- Change the numbers so that the mean is 10 times as large, 17 times as large, 3.9 times as large, 0.3 times as large, times as large.
Hey presto! A bog-standard two minute question (which is still important) has become an interesting 10 minute task.
Because I practise doing this regularly, I came up with these questions in just five minutes. (This is in contrast to writing this blog, which took me much longer because I don’t do it very often!)
I’ll leave you with a couple of missing digits problems in case long division is no longer providing you with a challenge. Before that, my favourite depth question (unashamedly stolen from an NQT I used to mentor) is:
“Show me three ways to…”
This is great because:
- It’s tough – one way to solve a problem is easy, two ways is usually ok, but three ways is usually pretty hard!
- It builds flexibility – ‘oh I could do THIS’ rather than ‘I have to follow a set procedure’ and it identifies and reinforces connections between different techniques and areas of mathematics.
- It’s easy to use as a teacher – just say ‘show me three different ways’.
Make sure you’ve done the maths yourself first though!