Beware the Ends of Branches

This post looks at certain topics which are often neglected when it comes to retrieval, why that happens, and what can be done about it.

Let’s start with a quick question. What seemingly accessible topics in your subject do students never seem to be able to remember?

In my subject, this is an easy one. Ask any maths teacher and their list will most likely include:

  • Loci
  • Constructions
  • Congruency
  • Simultaneous Equations
  • Cumulative Frequency Graphs
  • Averages from a table

Retrieval

Retrieval, retrieval, retrieval. It is a necessary part of learning (remembering?). Learning, not just performing, does not happen when something is seen once and then never again. You need introducing to something, you need to almost forget it, then you need to rescue it from your mind before it fades away just in time to keep it in your head for that little while longer.

In mathematics, some topics are naturally retrieved. With no effort from the teacher they will recur again and again. If we take solving two-step equations as an example, once taught as a discrete topic, it will rear it’s cheeky head when students move onto:

  • Forming and solving equations
  • More complex linear equations e.g. three step or with unknowns on both sides
  • Simultaneous equations
  • Quadratic equations
  • …

If we take basic angle facts, these will reappear when moving onto:

  • Angles in parallel lines
  • Forming and solving equations
  • Certain ratio problems
  • Angles in polygons
  • Circle theorems
  • Bearings
  • …

Why does this matter?

Retrieval, retrieval, retrieval. It’s just so important in embedding content, it means that the ideas above that recur are more likely to be learnt (remembered?).

A Curriculum Tree

If we imagine a tree of maths knowledge representing a curriculum. In this tree, the core ideas which appear everywhere form part of the sturdy trunk. Let’s imagine branches grow from pre-requisite materials, these more fundamental ideas mentioned above would form some of the more solid parts which would have many offshoots. As we track any branch to its end, you would find a topic that is not needed for any other topic for that curriculum.

The closer a topic is to the trunk the more naturally it will be retrieved. The topics at the end of the branches, however, are not going to naturally be retrieved whilst going through a curriculum. This is a problem. Why? Because retrieval, retrieval, retrieval, is so important to remembering (learning?).

Ends of the Branch

Because of how important retrieval, retrieval, retrieval is to learning (remembering?) we need to pay special attention to the ends of the branches. These will not recur naturally so we must find ways to force them into the curriculum. Most schools in the UK these days achieve this at the start of the lesson with some sort of retrieval activity. I think this is a great idea. Without revisiting content regularly there is a risk of it not being remembered and the start of the lessons seems a better place than most in which to do this.

It follows (if you believe all of the above) that the topics at the ends of the branch need special care and attention when designing these retrieval tasks. Beware the end of branches!

This all seems simple enough. Problem solved, right? Well, hold up a second.

There are some topics that, for various reasons, just complicate things.

I see hundreds of lessons a year and am yet to see a retrieval starter in which students need to recall loci or constructions, and very few where they retrieve cumulative frequency graphs, averages from a table or simultaneous equations etc. Does that list look familiar? (Don’t worry if you’ve forgotten it, we haven’t retrieved it yet), but it’s the list that we started this post with.

There is an uncanny crossover between what I do not see retrieved and the seemingly accessible topics that teachers will proport students struggle to remember. Could it be that it’s because students are not asked to retrieve this content as often? I would say so.

It’s perfectly understandable that this happens though. Nobody wants to spend the beginning of their lesson retrieving loci or constructions because it will inevitably lead to chaos as students do not have the necessary equipment.

Cumulative frequency graphs and averages from a table questions will require printing which is logistical barrier.

Simultaneous equations and congruency questions often require so much writing they may not fit into a snappy 5-minute starter routine your school so heavily insists on.

Solutions

If you or your department uses starts of lessons in this way then please beware the ends of branches. Not all of them, but those which require that little extra consideration. These can be easy wins but without putting in that extra effort every now and then to ensure students retrieve these topics, they are doomed to be forgotten (not learnt?).

Remember:

  • Take time to mark out explicitly when teachers should be reviewing these
  • Have some pre-prepared printable sheets ready
  • Have a class-set of equipment on standby for certain weeks
  • Be ready to (justifiably) take that little bit longer on the starter

Without taking special care, some branches will never be able to blossom.

I’m always interested in what people make of this so please feel free to comment with thoughts, questions or incomplete musings. Follow this or my Twitter account Teach_Solutions for similar content in the future. Also, check out the rest of this site, there’s some good stuff knocking about the place.

Leave a comment