# What were they thinking??

I do an increasing amount of work in primary schools and with primary school trainees developing maths subject knowledge and pedagogy. Over the last few years I’ve managed to blow my own mind many times just imagining the beginnings of concepts being developed in children. At secondary level (11+), most concepts at their basic root are pretty much embedded and understood or at least taken as ‘the way it is’. For example, a square is called a square, and the number after ten is called eleven. When I wrote the book ‘Yes, But Why?’, I really enjoyed putting myself in the position of a student being told all these strange and mystical maths facts and I tried my best to explain them to make them make sense. At a younger age, concepts must be even more mind boggling at first. An example I’ve used before is counting.

We count in base-10 (ie we use 10 symbols (0-9) and then start combining them. Imagine learning this for the first time, and getting to the number 9, then being told we start to use them all again. It’s weird right? And why do we use ten symbols? Why not five, or sixty? Does everyone count using the same system? (spoiler: no). These are thoughts we as adults probably don’t have, but they fascinate me. Who decided to call a square a square and not a quadrangle or a a tetragon? What were they thinking? Why are we using 360 degrees, and why are they called degrees? Where do these weird mathematical words come from? We’re used to using terms like polygon and acute angle, but these words are weird! “I’m going to find the mean of this data set”. The what?? Who decided to call it that? I “can’t subtract 3 from 1 so I need to borrow…” – you’re doing what now?? Can I borrow from elsewhere? Can I just move everything around? Also who decided that we should stack numbers like that to add or subtract them? What were they thinking? How did that idea become so popular?

Trying to think ‘like they do’ is a good first step into empathising with how abstract even the little details can be in mathematics, and how easy it is to misunderstand or misinterpret things. Furthermore, it’s a great exercise in testing your own subject knowledge, and figuring out what might be beneficial to incorporate into your own teaching and explanations to prioritise what must be prioritised – making maths make sense.

# Scaffolding

I saw an interesting lesson about ratios with a low achieving group last week which got me thinking about scaffold. The students had successfully been able to answer a number of questions in the form “A and B share £ in a ratio of x:y, how much do they each have?” and the teacher wanted to move on to a slightly more difficult variation in the form “A and B share £ in a ratio of x:y, if A has £, how much does B have?”.

For clarity, we’ll look at one specific example:

“Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?”

I’ve carefully chosen £420 here as it’s a nice round number and it fits into a number of potential misconceptions nicely too (hint, it’s a multiple of 4, 3, and 7). But this is less about that and more about ways to scaffold… so…moving on…

The students initially struggled with adapting to the new type of question, and i scribbled some notes about how I should write about how you could scaffold it. So here we are.

This is an exhaustive list, and I wouldn’t recommend doing *all* of these ideas in a single lesson necessarily. This is intended more as an insight into ways that may be necessary to help scaffold tasks to make them accessible.

The aim here is two-fold: make the maths understandable/accessible (usually by stripping it down), but also get to a point where the question as written is approachable – ie don’t strip it down without building it back up otherwise you’re missing the point.

Finally, there’s no specific ordering of ideas here necessarily.

Back to the question:

Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?

The first thing to note is that there’s a lot of text there, and two separate pieces of information that are particularly useful. A simple tactic here might be to simply break up the text:

• Ben and Andrew share in a ratio of 4:3.
• If Ben has £420 how much does Andrew have?

It may seem trivial, but you’re highlighting the two components to the question, and reducing/eliminating the need for the student to be able to discern the two things on their own (remember that your eventual aim is to get them to do this themselves).

You could further the support by adding a suitable diagram for them to complete:

• Ben and Andrew share in a ratio of 4:3.
• Draw a suitable diagram to represent this ratio
• If Ben has £420 how much does Andrew have?

Note the subtle difference below, the diagram has even more support in that the names are filled in on the diagram:

And if you’re concerned that they might not be able to draw the diagram correctly independently, then you could potentially ask a question like this instead:

Note that the scenario I gave was that the students could already confidently answer some ratio questions, so it’s likely that they can already draw suitable diagrams based on the previous types of question they were answering.

Below is a slightly different approach where the assumption is that they can draw the diagram, but that they might confuse the two names / ratios

Below is an example of less scaffold, in that the question has not been split up, but there is still guidance which could be considered more general to this type of problem:

• Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?
• Task 1: Underline the ratio
• Task 2: Draw a diagram to represent the ratio

Note that the two steps don’t solve the problem, they’re just a starting point. By underlining the ratio you’re getting students to draw attention to the important information instead of getting lost in all the words or rejecting the question because it’s “wordy” (recall we’re talking about low achieving students, not top set!).

Two more variations are below:

Again, the difference is subtle, and you know your students. Sometimes a slight change can be all that’s needed.

Now consider the question below:

• Ben and Andrew share £420 in a ratio of 4:3. How much does Andrew have?
• Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?
• Task 1: How are these questions different?

Here the shift is more to help students recognise how the new style of question is different. In Task 1 there isn’t really any maths to do, it’s more of a literature exercise – but it’s important. Without these discussions it’s likely the students will jump into question 2 using a question 1 approach.

Below we have assumed that the class are competent with determining the right diagram and drawing it accurately, so we move onto the second (new bit) part of the question:

I like this multiple choice element, which helps identify misconceptions about what the diagram is telling us, and where it’s telling it. Note that I’ve highlighted the part of the question we’re looking at with a different colour, to avoid anyone thinking we’re still talking about the first bit of the question. Note also that we’re giving away a bit here, in that we’ve inferred that one of A,B or C represents £420.

You can take that bit of help out quite easily:

Another approach to help students identify “what bit represents what” is to correctly or incorrectly complete it like this:

Again, note that we’ve given even more support in that we’ve specifically stated it’s wrong. Hence below is slightly less scaffolded:

Indeed, you could go the whole 9 yards and do something like this:

Note again that the explicit mention of how those numbers are derived is support you can add or take away.

Moving onto the final step in the question, students are likely to assume they have finished simply because they have reached a kind of end to part of it:

More structure:

Now remember that the ultimate aim is to take all the scaffold away eventually. When that’s done is of course at the discretion of the teacher. It may be that scaffold like the examples above only occurs in the teaching part, or it may be that independent questions have some of these kind of structures in place, but eventually you’ll want to move back to how the question was originally presented, something like this:

And then, remove the guidance altogether.

So there you have it. A brief guide to scaffolding a task in various ways. You can be as explicit / implicit as you like, but remember it’s about what makes it accessible to them, not how easy you can make it – and always aim to get to the point where the question as presented originally is approachable.