Chunk-based theory

This teacher training course is made of 5 topics, this topic: chunk-based theory, is made of 6 layers and is suitable for both teachers and managers within a school.

The teacher may read about each layer here and if desired or required can use the timely practice app to embed the course into their long-term memory.

(1) symptoms of working memory overload

There are signs of learners with small working memory capacities which we can glean from observation over time: https://thepsychologist.bps.org.uk/volume-21/edition-5/working-memory-classroom

There are 3 signs of working memory overload which we can witness as they happen - these can happen to any learners - even those with average or above average working memory capacities:

  • muddling methods

  • missing steps

  • giving up

Once we recognise each as a possible working memory overload response, then we are best placed to minimise it.

Let’s look at each of these in terms of maths learning.

muddling methods, very popular methods to muddle in the maths classroom are

  • calculating area versus perimeter,

  • median versus mode versus mean versus range,

  • column methods for addition versus subtraction.

The first two may be more because we usually teach them at the same time, rather than because they are naturally muddled. The third example of muddling column methods, comes I think, from the fact that all standard methods - have high working memory needs. I would say that column methods for addition and subtraction, long multiplication and division are paper saving but working memory heavy.

We are more likely to witness muddling methods when the learner is trying to recall learning, rather than when engaged in new learning. The other two symptoms missing steps and giving up may be witnessed during learning or when recalling learning.

missing steps: often happens when chunks in long-term memory are incomplete or when the working memory is overloaded by applying 2 methods together e.g.

  • how to calculate the range + how to subtract,

  • interpreting a stem and leaf + using a key which requires place value adjustment,

  • term to term rule for sequences + counting on the missing numbers,

  • expand: multiply in algebra + simplify by addition/subtraction + formal algebraic notation,

  • pictograms: using scale factor + calculating multiples,

  • not following the teacher’s instructions: too many instructions, hence some will be missed out.

As much as possible we need to improve learners confidence to get their workings on to paper - to reduce working memory load - but we may be working against years of shame for using non-formal methods, so gently does it.

giving up: there are 2 kinds of giving up we may witness

  • working memory overload - as it happens: where the learner is trying to calculate, but gets stuck, so uses an informal method, but gets stuck and so forth - so what looks from the outside as day dreaming or fake thinking may actually be overwhelmed thinking,

  • shame avoidance - the learner would rather give up before they begin and console themselves that they “didn’t try”, “you didn’t teach us” or “this won’t be useful in my adult life” rather than try and fail. This behaviour can be seen as a way of dealing with cumulative failure. If we make learning maths too stressful, then our learners will be primed for forgetting the uncomfortable feelings that the lesson induced, as well as any maths they may have learned during that lesson. See also maths anxiety.

The first type of giving up is the easier to work with than the second, but both types are what timely practice was built to improve.

(2) build + improve chunks/triggers in Long-Term-Memory

timely practice repurposes the research on how experts learn to ensure that embedding learning is easier for low attaining maths learners.

Experts use deliberate practice to build up complex chunks of skills and knowledge in their long-term memory.

Chunk-based theory says that chunks are built of two parts:

  1. the trigger “use me I think I’ll be useful for a problem like this”

  2. the rest of the chunk which holds the knowledge and skills.

Experts use chunks in place of working memory - hence they can appear to have super-human working memory capacities - in their field of expertise.

It takes time and appropriate practice to build chunks in long-term memory.

Chunks with slots for variables are called templates.

Mental schema are another name for chunks or templates.

Consolidation is the process whereby a brain state in active or working memory is stored in long-term memory. This process modifies synapses on the dendrites of neurones. After retrieval of the memory, a similar process, called reconsolidation occurs whereby the old memory is altered and replaced by the new memory.

Multiple retrievals and reconsolidations may be needed to build an accurate chunk. It seems we are not prompted to reconsolidate

  • if we can easily recall our learning - hence overlearning is far less effective than retrieval practice,

  • by activities which induce uncomfortable feelings - what we might have learned, is not laid down in long term memory - perhaps to preserve our self esteem

Both consolidation and reconsolidation happen during sleep, so we can’t possibly know what a learner has learned during a lesson. We must wait for at least one nights sleep, to find out what has become learning and what has not.

It seems that if recall is too easy, reconsolidation won’t happen at all or make a perceivable change in long term memory. This fits with Bjork’s desirable difficulties.

It seems that by its very definition reconsolidation can’t happen at the end of the lesson where the skill is taught, because the learner hasn’t had a sleep so hasn’t even consolidated the learning.

(3) overcoming low attaining learners' double whammy

Chunks can reduce the working memory requirement to solve problems and process information and to learn. However

Learners with smaller working memory capacities are less likely to build chunks in long-term memory after the lesson than their peers.

Learners with smaller working memory capacities are more dependent on chunks than their peers to process the content of lessons.

I call this a double whammy, because learners with smaller working memory capacities are doubly disadvantaged - and often go on to become low attaining learners. I would say the best thing we can do as teachers is to teach low attaining learners in a smaller working memory friendly manner.

(4) feedback is best after a nights sleep + how to fix incomplete and incorrect chunks

Contrary to our expectations, feedback is better given after one nights sleep, than directly after an error. It seems that if we give feedback on the day of the error, we may not be as effectively triggering reconsolidation - see (2) above - that is we are not as effectively triggering the brain to change chunks in long-term memory. On the other hand, if we leave feedback for too many days, then feedback is not as effective as it could be either.

I wish I could say that is why timely practice expects the teacher to assess assignments between one lesson and the next - and only then give feedback, but these fundamental decisions were made:

  • to reduce the reliance of learners on having one computing device per learner and reliable access to wifi in every lesson,

  • to get learners to write by hand their workings and diagrams, just as they must in a test and quite possibly because it’s more effective at embedding learning.

There are number of general problems that feedback may need to overcome.

  1. The learner has built an incomplete chunk in long-term memory - despite timely practice layers being small and therefore easier to learn, sometimes the learner will need more support to build a chunk - the teacher, via feedback-dialogue, should work with the learner to find what is missing and help the learner fix it. See also (5) fading scaffolding, for more about this. We recommend assessing rather than marking of assignments, to assist with this. With marking, the teacher might write a note to the learner, showing the missing bits, but with feedback-dialogue, we help the learner add the missing bits to the chunk, ideally via questioning rather than telling.

  2. The learner has not replaced/adapted an incorrect chunk built some time ago, with a new/adapted chunk in long-term memory. This is different from 1. in that the chunk to do the old incorrect method hasn’t been overwritten, despite perhaps a new chunk being built in the lesson. So here we are working on changing the trigger i.e. the learner choosing the new correct chunk, rather than the old incorrect one. The best way to do this, is to offer a reason why the old incorrect one is incorrect, that chimes with the learner’s understanding.

  3. The learner is still reliant on some of the “unacknowledged scaffolding'' of the lesson e.g. placement of workings out on the page or use of a diagram etc. See (5) fading scaffolding, below for more about this.

  4. The learner has misread the question or poorly applied their numeracy skills when answering the question. This is likely to be due to working memory overload - all learners, even the most able A level maths learners experience it. As they are learning and working through something new and hard, they are unable to accurately apply skills which are usually easy for them. I sometimes describe this effect to students as their brain isn’t very good at easy thinking and hard thinking at the same time. The best we can offer learners as they practise, is that they can look through their workings out for accuracy periodically. Sometimes I suggest they write “check for accuracy” on the answer line, as an aide memoire, for when they think they have solved the problem. We need to encourage learners to realise that making “silly mistakes” is often a sign of hard learning going on, not a sign that “they can’t even do the easy maths”.

(5) fading scaffolding

More about fading scaffolding is found in (4) feedback is best after a nights sleep + how to fix incomplete and incorrect chunks above.

The difference between being able to answer questions in a lesson, where all the questions require the same, recently learned skill, and being able to answer the interleaved retrieval practice questions within the learners assignment is a little like the difference between swimming with and without a float. As teachers we are often unaware of how much scaffolding is in the classroom when we are teaching a topic e.g. notes on whiteboard, vocabulary fresh in learner’s minds etc. Additionally, in the lesson where teaching occurs, learners don’t need to use the triggers for their chunks in long-term memory, they just need to remember the topic of the lesson. Almost all the practice questions (except of course those in their timely practice assignment) that learners will do in the lesson, will be on the topic of the lesson.

Once learners begin doing retrieval practice questions in their timely practice assignment, they have to

  • read the question and decide which skills to apply and

  • recall the complete process they learned in a previous lesson,

so it is not surprising that sometimes we will see occasions where teaching hasn’t become embedded learning (yet).

To help the learner recall the scaffolding of the lesson for themselves, we can use prompts to trigger the chunk e.g.

  • “what diagram will I ask you to draw?”

  • “where on the page, would it be best to show on the side workings out?”

When we ask questions like this we help nudge the learner to replace the external scaffolding of lessons with their own internal scaffolding - i.e. a better/bigger/more complete chunk in long term memory. It is often clear, that that is what we are doing e.g. as as soon as we say “what diagram”, we see the “aha” look and the learner wants us to go away and leave them to get on, they know what they are doing. The diagram, often already has a chunk attached to it, in the learner’s long term memory.

To help the learner recall all the steps of a process, is slightly different, we aren’t trying to get the learner to recall a fully finished chunk, rather we might be trying to

  • get the learner to join 2 or more chunks together e.g. to write 7% as a decimal, we are trying to get the learner to recall and join the chunk of how to write 7% as a fraction and the chunk that knows that the fraction 7/100 also means 7 ÷ 100 and the chunk that knows how to divide a number (without a decimal point) by 100.

  • to add something new on to the end of an existing chunk e.g. to add on to the chunk for finding the median of an odd number of data items, the extra steps to find the median of an even number of data items.

We have to accept that selecting best learned later and hoping that “in class teaching and practice” in the next spiral of the curriculum, is unlikely to solve the problem - if the problem is - the learner could answer practice questions in the lesson they were taught, but can’t recall everything the next lesson. As it is clear that rather than being unable to apply the process but being unable to recall to recall the process which is the problem.

  • Hence, we are unlikely to fix the recall problem, with another lesson - whether the lesson is next term or next year - when we tell the learner what to do (unless, there are parts of the process that are not yet mastered).

  • We are however, much more likely to fix the recall problem, by a few quick feedback-dialogue sessions, in the next few lessons, especially if we see what the learner can already recall and then help them add a bit more to the process or join existing processes together.

In our development of this version of timely practice we have created many scaffold pair layers, one of which carries with it some of the scaffolding of the lesson. These make it easier for us to “reduce the amount of new” from one layer to the next. Here are 2 examples of scaffold pair layers.

given - sign layers 3 and 4

given - sign layers 9 and 10

At the moment, November 2022, the app can’t automatically swap between scaffold pair layers - but we plan to do this in the future.

Not all layers, which have a scaffolded layer planned, are written yet. This is the main reason why some topics e.g. frequency table has gaps between some layers.

(6) problem solving + combining chunks in novel ways

In (5) fading scaffolding, an example of the need to

get the learner to join 2 or more chunks together e.g. to write 7% as a decimal, we are trying to get the learner to recall and join the chunk of how to write 7% as a fraction and the chunk that knows that the fraction 7/100 means 7 ÷ 100 and the chunk that knows how to divide a number (without a decimal point) by 100.

was given. I don’t think many teachers would count this as problem solving, because we are requiring learners to use a number of skills to perform a standard problem.

Problem solving, requires more

  • it could be what many of us might see as too simple to be problem solving: answering a word problem, which we at timely practice classify as secret x sign, that requires the learner to realise that the problem will be solved by multiplication,

  • it could be a more complicated word problem: e.g. requiring both multiplication and addition, which I think most teachers would classify as problem solving,

  • it needn’t be a word problem e.g. the diagram below could provide simple or complex problems:

2 of the 3 semi circles are the same size
  1. it could be that the learner is told the diameter of the larger semi-circle and asked to find the diameter of the smaller semi-circles,

  2. it could be the learner is told the diameter of the shape and asked to find the area of the shape,

  3. it could be the learner is told the diameter of the shape and the price of 1 cm2 of gold leaf, and asked to find out the cost of covering the shape with gold leaf.

These problems are hardly novel, but I would say 2. and 3. require genuine problem solving skills: they require

  • the chunk for what diameter/radius mean and/or how to work out diameter from radius or v.v.

  • the chunk to work out the the area of a circle,

  • and combining these chunks in novel ways.

What we know from research on cumulative practice is that learners are much more likely to be able to solve problems, if they have mastered all the pre requisites. So timely practice provides a layer shape problems NC layer 2, to answer questions like 1. in the example above. I doubt that such a question will be asked in a GCSE paper, but the layer provides consolidation of the meaning of radius and diameter. Mastery of this layer and mastery of shape skills YC layer 6, how to find the area of a circle, might make solving problems like 2. within the learners grasp. Whereas I think, most maths teachers, with experience of teaching maths learners below the lower quartile, would be doubtful if such learners would be able to solve problems like 2. and 3. in a lesson, but certainly would not expect them to be able to solve such problems weeks after the lesson.

The timely practice way to teach problem solving, is through cumulative practice.

Disclaimer Many low attaining learners, come to timely practice late in their schooling. Such learners are unlikely to be able to solve problems like 2. and 3. because they simply run out of time before they take their GCSE.