Category Archives: For Maths Teachers

Present new material in small steps with student practice after each step: How’s it look?

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The second recommendation in Rosenshine’s ‘Principles of Instruction is “Present new material in small steps with student practice after each step”. The basis for this recommendation is the fact that working memory is limited and, for learning to occur it’s important to avoid overloading working memory. But that isn’t the focus of this post. In this post I just wanted to share what ‘new material in small steps with student practice after each step’ can look like in the classroom.

As a rule of thumb, the longer a teacher talks for the more likely they are delivering sufficient information to overload their students’ working memory. As I reflected upon this point, prompted by Craig Barton’s in-depth interview with Kris Boulton recently, I found myself thinking, ‘I wonder how long I talk for?’ It was time to collect some data.

Next lesson I split my notebook into three columns ‘explain’, ‘student work’, and ‘check solution’ (I always teach my maths lessons in an ‘I do’ then ‘You do’ format, then go over the solutions as a class), then I got to recording! First class I got distracted and fell off the timing bandwagon (first half of the page) but second class I remembered to stay on task and that whole class (90 mins) is recorded in the image below (red box).

To set the scene, I wanted students to be able to answer the exam question presented by the end of the lesson. This required them to be able to go from a transition diagram and an initial state matrix to the result after multiple periods with or without the addition of extra units each period, as well as determining the result of such transitions ‘in the long run’, and working backwards in such a relation. I split this up into the following sub-steps for the purposes of instruction.

  • Constructing a transition matrix from a transition diagram.
  • Applying a transition diagram to interpret a transition
  • Applying a transition matrix to interpret change after one transition
  • Understanding transition matrices as recurrent relations (And results after multiple periods with a formula)
  • ‘In the long term’: Steady state solutions to Transition matrices
  • Results after multiple periods (using brute force, that means with a calculator)
  • Transition matrix modelling when the total number of units changes.
  • Working backwards in matrix multiplications


The astute observer will note that the total time adds up to about 60 mins. The additional time was taken up with approx. 20 mins of revising previous content and 10 mins talking about an upcoming assessment and doing a ‘brain break’.

Below is the lesson as I presented it, with the timing for each segment added in italics (images weren’t in the original as students had all questions in front of them. I added them for readers here)

I found it really valuable to look at the timing of my lessons in this level of detail. I’d love to know if it’s prompted any similar reflections for you.


Rosenshine, B. (2012). Principles of Instruction: Research-Based Strategies That All Teachers Should Know. American Educator, 36(1), 12.


Sweller’s Goal Free Effect… giving it a go.

Thanks to a recent tweet by Dylan Wiliam, and a great article that it linked to by Michael Pershan, I gained a fuller understanding of a cognitive effect that I’ve been exploring recently (see this paper), the ‘goal free effect’.

Discovered by John Sweller, it essentially posits that explicitly trying to solve a problem can result in a lot of ‘attention’ or ‘working memory’ (see here for a discussion of which term to use) being expended in the search process, limiting (or eliminating) the working memory available for ‘learning’ from the actual task. The result is that the problem gets solved, but the problem solver fails to make any generalisations from the solution and won’t be able to necessarily do it again in future.

It doesn’t come across as a a particularly complex theory, but what I’ve been trying to work out is how to make it work in a classroom. I read Pershan’s post but was keen to know more about linking the goal free approach to explicit learning intentions that the teacher has for the lesson (we discuss that here if you’d a bit more detail on this chat).

Sometimes it takes trying something out to get your head around it, and I was determined to do so. This week I encountered a question that I wanted my students to be able to solve, and I thought the goal free effect might be relevant. Here’s the question (see part b):

Question to use with the goal free effect

I recognised that there was a danger here. This was a relatively open question and I anticipated that several of my students would find it difficult. I could anticipate that many of them would just stare at the table without making connections and then after some work time and a few prompts I’d show a solution (or they’d find it themselves in the resource) but, because they’d been so solution focussed along the way, they’d just write the provided solution down and try to memorise it (the provided solution just focussed on the trend for 19 years and under) and fail to see all of the associations that they could have pointed out in the table.

What I did instead was try out Sweller’s theory.

I clipped out the table and showed it by itself on the whiteboard with my projector. I then asked ‘Look at this table… What can you tell me by looking at it? Do you notice any patterns?’

I also gave the following hint: ‘Focus on one row ( ← a row goes like this → ) at a time’.

We then shared as a class and it was an incredibly rich discussion. What I hadn’t anticipated was how asking such a question reduced the barrier to participation for students. I had students point out the patters for each of the age groups, but I also had one student say ‘The years go up in 10s’ as well as another similarly volunteer that ‘The years all end in 6’, this was in addition to associations being found between the year of first marriage and age of first marriage for each of the age categories in the table.

I then gave each student a half sheet of A4 paper and got them to put into words their association (I’d identified from the discussion that students were struggling to put their thoughts into formal mathematics terms, so wanted them to make these descriptions less transient by eliciting a written response) and collected up these bits. I read some out and, as a group, we identified what it was that made the strong ones strong. I hadn’t anticipated this at all, but we ended up making a template for answering these such questions, here it is:

Template from goal free effect activity

For me this was an incredible experience. We’d made it all the way from an open question to a generalisation, and scaffolded literacy along the way too (I work in a very low SES school with a large English as Additional Language student base, literacy needs are a constant in all classes), something I’d failed to anticipate in my planning.

In carrying out this activity I managed to get a much deeper understanding of how the goal free effect can work, and how it can be tied into a generalisation directly in line with my learning intention for this segment of the lesson (FYI, the explicit learning intention was for them to be able to identify associations from a two-way contingency table then describe the association and back up their claim with data from the table). In future cases, especially when there’s a lot going on in a diagram (see ‘split attention effect’ on bottom left of page 6 in this paper) I’ll definitely have the goal free effect in the back of my mind as one option in my teacher toolbox.


Top 9 Math Question Websites and Bloggers for deeper thinking (Shane Loader)

A big thanks to Shane Loader for the following recommendations.

Shane uses resources from the following sites to promote “problem solving and reasoning skills and the flexible use of mathematical knowledge in students”.

Enjoy! I’ve found them incredible!

The resources:

Which one doesn’t belong                 

Open Middle                                           

Visual Patterns                                        

Robert Kaplinsky’s Tasks                     

Andrew Stadel’s Three act tasks     

Dan Meyer’s three act tasks             

Yummy Math                                           

Estimation 180                                         

Would you rather                                  

Edit: For a maths blogosphere specific search engine, check out the link here.