NW3 Maths Hub Conference

This Wednesday I was lucky enough to deliver the closing keynote to the Wigan NW3 Hub Conference at Haydock Racecourse. I absolutely loved the chance to mirror the development of an operation through a counters game, before exploring the importance of making sense of mathematics through the power of multiple interpretations of a concept.

The slides from my session are here (including the correct formula - well I hope so anyway!).

Thanks to Lindsay Porter for inviting me to speak (and giving my a lift back to the station) and to Jen for picking me up from Bryn station.

SEND Conference from LIME/Maths Hubs

On Monday 24th June I had the privilege of presenting the closing session at an excellent event hosted at the Ashton-on-Mersey school. This was primarily for teachers of pupils with SEND. The main focus was on the use of manipulatives to support mathematical understanding - a personal favourite topic of mine.

Many delegates suggested they would find the slides useful, so I have made them available for download here.

I should give a shout out and offer thanks to Louise Needham for asking me to speak at this fantastic event.

Putting the "Theory" into Cognitive Load Theory

These days we are hearing a lot about Cognitive Load Theory. But what does this actually mean? Well to understand this it is worth reminding ourselves about what it means to be a theory in science.

A lot of people see the idea of a "theory" as something that is somewhat uncertain. This is often the use in everyday language - if someone has a "theory" about something, it often means they have no more than a vaguely plausible explanation for it.

A scientific theory is different though (or at least a good one is). A good scientific theory should broadly aim to do two things:

1) Explain observed phenomena

2) Predict the outcomes of other observed phenomena

This is what Cognitive Load Theory tries to do. It tries to explain phenomena about how/when the brain forms memories that have been observed, and predicts what might happens in certain circumstances. For example, it has been observed that people find it difficult to remember content if they are reading text at the same time as someone is talking. Typically people in this situation will not be able to answer questions about either the text or the content of the speech. CLT explains this by suggesting that the brain processes text in the same way as speech (in a way, you "hear" the words in your head) and that the brain only has one "channel" for processing auditory input. Trying to process two inputs through your "phonological loop" results in cognitive overload.

So what happens when a prediction goes wrong? What happens when CLT predicts a different outcome? Well the same as what happens when any other scientific theory predicts something incorrectly - either the theory is modified to include the new observation, or if it can't be modified sufficiently then it is deemed incorrect. However, incorrect theories can still be useful. A prime example of this is Newton's theory of gravitation.

Newton's theory of gravitation is wrong. It definitely doesn't adequately explain how gravity works in all cases. This was known in the 1800s, as Newton's theory of gravitation was slightly out in predicting the correct orbit of the planet Mercury. Einstein's general relativity is a better model. Its predictions are more accurate, and more applicable. However, in most cases, Newton's theory is still used. Why? Because it is much simpler. The equations that accompany Einstein's general relativity are absurdly complicated. If you are talking about black holes, or getting close to massive bodies in the universe, they are essential. But for most situations, the equations associated with Newton's theory do just fine. They predict to a high level of accuracy the gravitational forces between bodies. Newton's theory was used to put man on the moon.

So what does this mean? Well if we apply the same sort of ideas to Cognitive Load Theory, what it means is that CLT may well make incorrect predictions, particularly in extreme cases, but that doesn't necessarily mean that the other predictions it makes are automatically wrong, or that they can't be useful. But it also means that if you are going to try and apply the ideas within Cognitive Load Theory then it might be useful to remember the following:

1) CLT may well not a complete theory of cognition, and it may well produce incorrect predictions. This doesn't make it worthless.

2) If you are applying CLT, make sure you read information about the studies that supported aspects of the theory. This will give you a greater appreciation for how useful/accurate its predictions might be for your context.

3) Cognitive Load Theory may well support in your pupils converting more of what you teach into long term memory, but that also means you have to make sure that the memories you are getting your pupils to form are the right memories. CLT can't tell you how to teach the content of your subject so that the connections between topics become apparent - that is part of the knowledge/skill (contentious!) of the teacher.

Research Ed Rugby - Mathematics Teaching for Mastery Using Rosenshine's Principles and Cognitive Science

This Saturday I attended the truly excellent ResearchEd Rugby. Despite all sorts of problems with trains from London to Rugby, Jude Hunton (organiser) got a fantastic line up of speakers together for sessions on Research, Leadership, Maths, English, Science among others.

As part of the maths strand I was speaking in the afternoon. The thesis of my talk was that the NCETM approaches outlined in their Teaching for Mastery program, Rosenshine's Principles of Instruction, and some of the effects proposed by Cognitive Load Theory are the same ideas, discussed using different language.

It is worth reminding ourselves about how these three came into being. The NCETM ideas of Teaching for Mastery came from looking at "high performing jurisdictions" and their practice, as well as the research that underpins their approaches, and suggested what might have the most impact in mathematics education in this country.

Barak Rosenshine derived his Principles of Instruction by examining individual high performing teachers, and the common practice that they share. This of course was not specific to mathematics teaching.

Cognitive Load Theory is a theory for how the brain forms memory, and things that support forming of memory, based on experimental data. Of course, the point of a theory is that it explains observed phenomena, and predicts the outcomes of future phenomena. So CTL aims to explain things that have been observed about memory formation, and predict what might help memory formation in the future.

The three "competing" theories can be shown using these three images:

I contend that many of the ideas from these three sources actually significantly overlap, and in some cases are indistinguishable. Take this example from my presentation:

On the left hand side is a picture of the expression x² + 5x + 6, which then shows how this picture can be rearranged into a rectangle, which shows the factorisation. This can be physically shown using concrete or virtual manipulatives. Then there is an expression on the right hand side which is intended for the learner to factorise (given access to the concrete version of the manipulative).

The title for the slide was:

The proposition here is that this could be considered Modelling, if thinking about Rosenshine's principles, or it could be considered Representation if thinking about NCETM Teaching for Mastery  approaches, or could be considered the use of the Worked Example Effect if thinking about Cognitive Load Theory.

I highlighted several other examples throughout the session:

This activity is one that can be used to support developing Fluency, which is also an example of independent practice and also uses the Goal Free Effect.

This activity prompts Mathematical Thinking, it can be seen as using the Expertise Reversal Effect, and can be used to provide Scaffolding for Difficult Tasks. Note; if you are going to use this task, an important point is that learners should aim to make the minimum change possible from the starting shape in the middle.

The point of course is that these are not competing ideas at all. There is something to be gained from all of them, particularly where they actually say the same thing.

*Hat tips to the Learning Scientists and Oliver Caviglioli for the posters, Jonathan Hall and his website Mathsbot.com for the virtual manipulatives, Open Middle for the open box problem, and Professor John Mason for the Area/Perimeter activity.

*The full presentation can be downloaded by clicking this link.