And 40 divided by 20 is...

Why maths teachers cannot resist this type of question, and why it is a problem

Want free Wonde access to sync and manage pupil information on Eedi with one click? Get started on Eedi with ease by signing up here and we'll be in touch!

Once upon a time…

Recently, I observed a teacher, Mr L, modelling how to solve a pair of simultaneous equations. Let’s pick up his explanation midway through:

…the signs on the ys are different, so we need to add the two equations.

Tom, what is 15x + 5x?… yep, 20x.

The y’s cancel out to leave us with 20x = 40.

To solve, we need to divide both sides of the equation by 20.

Emma, what is 40 divided by 20?… yep, 2.

So, x equals 2…

Mr L gives himself a metaphorical pat on the back, safe in knowing his students are following his work example. His students smile back, concluding they must understand what is happening because they are getting Mr L’s questions correct.

When students tried to solve a pair of simultaneous equations independently, they quickly came unstuck. Neither they nor Mr L could not understand why,

What is going on here?

During the worked example, the teacher asked his students two questions:

1. What is 15x + 5x?

2. What is 40 divided by 20?

What is the purpose of these questions?

Are they to check students’ understanding of how to solve simultaneous questions? Definitely not, as zero knowledge of solving simultaneous equations is needed to answer them.

So, are they to check that students are paying attention to the model? Again, no. A student could have tuned out of the entire model, pricked up their ears as soon as their name was mentioned, and then answered the straightforward question they then heard. A proper Cold Call technique (question, pause, name) would reduce this possibility, but it is still not a proper check for listening.

So, what purpose do these questions serve?

They are there to convince everyone—teacher and student alike—that things are going well. Teachers convince themselves that their explanation makes sense as their students contribute to the solution. Students convince themselves they understand as they can answer the teachers’ questions.

So what is the solution?

Let’s start with something that isn’t the solution: asking students questions that check their understanding of how to solve a simultaneous equation. Students don’t know this yet as they have not been taught it. Sure, some students might be able to figure it out, given time and prompting. But what about the other students? How do they feel? How do they benefit? And what about the opportunity cost of the time this takes?

No, when modelling, two approaches are much more effective.

1. Don’t ask any questions. Assess relevant prior knowledge first. Then deliver a clear and concise explanation linking the things that students are already familiar with within the novel process you are demonstrating.

2. Only ask questions that check students are listening. And ask these questions for the new, vital part of the process, not the easy parts that anyone can get right without paying attention. Explain to students why you need to add these two equations, and then check if they can tell you. Divide both sides of the equation by 20, and then check if students can tell you where the new equation came from. Use techniques such as Explain, Frame, and Reframe to make these checks for listening more cognitively demanding.

Then, properly check for understanding by having students complete the We Do on mini-whiteboards using a Step-by-Step approach. Then, you will have real evidence of whether your explanation made sense.

Do you recognise the 40 divided by 20 questions in your teaching?

What do you agree with, and what have I missed?

Let me know in the comments below!

🏃🏻‍♂️ Before you go, have you…🏃🏻‍♂️

1. … checked out our incredible, brand-new, free resources from Eedi?

2. … read my latest Tips for Teachers newsletter about the relationship between interest and cognitive load?

3. … listened to my most recent podcast about the Do Now?

4. … considered purchasing one of my new 90-minute online CPD courses?

5. … read my Tips for Teachers book?

Thanks so much for reading and have a great week!

Craig