Variation Theory

Luckily for me, my father in law is also a Maths teacher, and a huge enthusiast for all things maths. Unfortunately, it also means that family dinners can become quite boring for the rest of the family as we will often gabble about ways of teaching exciting maths topics and excellent resources we have recently discovered.

He recently bought me a copy of Craig Barton’s book ‘How I wish I’d taught Maths’, and whilst I was already familiar with Mr Barton’s resources and podcasts, it was interesting to read about his theory behind the resources he shares and how he came to create and share these.

For many of you reading, none of these ideas may be new to you, but it has made me really think about my practice, my style of teaching and my thought process when planning sequences of lessons.

Barton talks a lot about cognitive load theory, and how much a student can store in their working memory, and how skills / ideas can be transferred from their working memory to their long term memory. He also talks about (something I am VERY guilty of) ‘great little activities’. Those activities you pull out every year when teaching a specific topic, that students enjoy, you enjoy delivering, that ignites their passion for maths (for that one lesson) that you are sure will enable them to remember the skill you have taught them forever and ever.

For example when teaching Pythagoras to bottom set Year 9, I’d prepared a huge right angled triangle with squares coming off each side, using masking tape on the floor of my room and moved all the chairs and tables away from the exciting demonstration. I had them physically lay out post it notes to completely fill two squares coming off the smaller sides in a right angle triangle. They then used the post it notes from the two squares to completely fill up the square coming off the side of the hypotenuse of the same triangle. One student filmed this and then we sped it up and made a cool-looking video demonstrating Pythagoras’ Theorem in action. We watched the video every lesson for the next few lessons to remind us how Pythagoras worked and I was certain that students would remember that  for the rest of their days. What they actually remembered was a ‘fun’ lesson in which they got to film stuff and play with post-it notes.

Post it notes 2

Whilst ‘fun’ activities can capture imagination, raise engagement and stick in a student’s memory, whether a student has actually learnt anything valuable that will be stored in their long-term memory is debatable: not impossible, but debatable.

Based on Barton’s theory, we should teach skills through guided practice, and vary our questions in a specific way to enable students to not only work out the answers accurately, but to enable them to notice and discuss changes in questions, use this to predict answers and then check their predictions using the skills they have been taught.

I would always use a worked example and a ‘you try’ approach before students were encouraged to fly the nest and fend for themselves, but these two examples were usually quite different, with little similarities in the numbers/ examples I chose. I would check understanding and then give a significant amount of time over to independent practice. Here is an example of some percentage questions that I would have given my class in the past, once I’d showed them how to calculate percentages of amounts. (taken from )

Mr Carter Maths

There’s some differentiation by task here, and I could decide who starts where. I may allow students to choose which section they start on, allowing progression to harder questions when they felt ready. I may insist that students complete at least 2 from each section, and have challenge resources ready for those that finished.

I do still find these questions useful, perhaps at the end of teaching this topic, for a homework task or as a revision task.  But now, the examples I’d use in lesson look more like this (taken from )

Variation Theory

As you can see, there is huge similarity from one to the next. I sometimes use Barton’s specific approach detailed here:

” 1.  I carry out the procedure using the Silent Teacher approach, hand-writing my worked solution on the board, prompting the students to reflect at key points in the process, asking themselves what did he do then? and what will he do next?

2. I then Narrate and Annotate, this time explicitly asking students questions for them to consider such as where did the 30 come from?and what does the 5 represent?Having given students time to reflect on each question, I will explain in my own words and then annotate the board. 

3. Students then copy this example down into their exercises books

4. Students then try a related problem on their own, discussing their solution with their neighbour once they have both finished:”

I then move onto the ‘intelligent practice’ set of questions, either going through these as a whole class or with students going through at their own pace, depending on what works best for that class.

Maths Sums

Whilst the questions look simpler, I would be ensuring students followed these three steps (from Mr Barton) when answering each question

  • Reflect – what has changed in this question to the previous one? What has stayed the same
  • Expect – based on their reflection and the answer to the previous question, what do students expect the answer to the current question to be?
  • Check – students can then check their expectation by carrying out the procedure in the manner modelled during the worked example

This approach doesn’t fit with every single topic, but for the topics it does, I can definitely say I have seen a greater impact since using this approach, with students more confident, more vocal and better at spotting patterns and predicting changes in answers. It also works well for all ability groups and all ages.

In terms of differentiation, students can progress through the questions faster, or start at a later point in the sequence of questions, they can prepare comments for any unexpected answers and try to dig out why an answer wasn’t what they expected it to be. For students who find the skills easy, it offers the opportunity to think about the underlying structures of the topic and build a deeper understanding.

Students could then be moved onto a bigger variety of questions, perhaps worded exam style questions, perhaps multiple choice ‘diagnostic questions’ or perhaps some ‘same surface different structure’ questions (as detailed in a previous ‘top tip’).

If you have any questions about Variation Theory or would like to see it in action, just send me an email!

Harriet Lambert, Maths


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s