going round in circles

Central to the Tiger Teachers’ model of cognitive science is the concept of cognitive load. Cognitive load refers to the amount of material that working memory is handling at any one time. It’s a concept introduced by John Sweller, a researcher frequently cited by the Tiger Teachers. Cognitive load is an important concept for education because human working memory capacity is very limited – we can think about only a handful of items at the same time. If students’ cognitive load is too high, they won’t be able to solve problems or will fail to learn some material.

I’ve had concerns about the Tiger Teachers’ interpretations of concepts from cognitive science, and about how they apply those concepts to their own learning, but until recently I hadn’t paid much attention to the way their students were being taught. I had little information about it for a start, and if it ‘worked’ for a particular group of teachers and students, I saw no reason to question it.

increasing cognitive load

The Michaela Community School recently blogged about solving problems involving circle theorems. Vince Ulam, a mathematician and maths teacher*, took issue with the diagrammatic representations of the problems.

The diagrams of the circles and triangles are clearly not accurate; they don’t claim to be. In an ensuing Twitter discussion, opinion was divided over whether or not the accuracy of diagrams mattered. Some people thought it didn’t matter if the diagrams were intended only as a representation of an algebraic or arithmetic problem. One teacher thought inaccurate diagrams would ensure the students didn’t measure angles or guess them.

The problem with the diagrams is not that they are imprecise – few people would quibble over a sketch diagram representing an angle of 28° that was actually 32°. It’s that they are so inaccurate as to be misleading. For example, there’s an obtuse angle that clearly isn’t obtuse, an angle of 71° is more acute than one of 28°, and a couple of isosceles triangles are scalene. As Vince points out, this makes it impossible for students to determine anything by inspection – an important feature of trigonometry. Diagrams with this level of inaccuracy also have implications for cognitive load, something that the Tiger Teachers are, rightly, keen to minimise.

My introduction to trigonometry at school was what the Tiger Teachers would probably describe as ‘traditional’. A sketch diagram illustrating a trigonometry problem was acceptable, but was expected to present a reasonably accurate representation of the problem. A diagram of an isosceles triangle might not be to scale, but it should be an isosceles triangle. An obtuse angle should be an obtuse angle, and an angle of 28° should not be larger than one of 71°.

Personally, I found some of the inaccurate diagrams so inaccurate as to be quite disconcerting. After all those years of trigonometry, the shapes of isosceles triangles, obtuse angles, and the relative sizes of angles of ~30° or ~70°, are burned into my brain, as the Tiger Teachers would no doubt expect them to be. So seeing a scalene triangle masquerading as an isosceles, an acute angle claiming to be 99°, and angles of 28° and 71° trading places, set up a somewhat unnerving Necker shift. In each case my brain started flipping between two contradictory representations; what the diagram was telling me and what the numbers were telling me.

It was the Stroop effect but with lines and numbers rather than letters and colours; and the Stroop effect increases cognitive load.  Even students accustomed to isosceles triangles not always looking like isosceles triangles would experience an increased cognitive load whilst looking at these diagrams, because they’ll have to process two competing representations; what their brain is telling them about the diagram and what it’s telling them about the numbers.  I had similar misgivings about the ‘CUDDLES’ approach used to teach French at Michaela.

CUDDLES and cognitive load

The ‘traditional’ approach to teaching foreign languages is to start with a bunch of simple nouns, adjectives and verbs, do a lot of rehearsal, and work up from there; that approach keeps cognitive load low from the get-go.   The Michaela approach seems to be to start with some complex language and break it down in a quasi-mathematical fashion involving underlining some letters, dotting others and telling stories about words.

Not only do students need to learn the words, what they represent and how French speakers use them, they have to learn a good deal of material extraneous to the language itself. I can see how the extraneous material acts as a belt-and-braces approach to ‘securing’ knowledge, but it must increase cognitive load because the students have to think about that as well as the language.

The Tiger Teacher’s approach to teaching is intriguing, but I still can’t figure out the underlying rationale; it certainly isn’t about reducing cognitive load.  Why does the Tiger Teachers’ approach to teaching matter?  Because now Nick Gibb is signed up to it, it will probably become educational policy, regardless of the validity of the evidence.

Note:  I resisted the temptation to call this post ‘non angeli sed anguli’.

*Amended from ‘maths teacher’ –  Old Andrew correctly pointed out that this was an assumption on my part. Vince Ulam assures me my assumption was correct.  I guess he should know.

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6 thoughts on “going round in circles

  1. Hi

    I had a quick look at the CUDDLES post and the following Knowledge Organisers, Quizzes and Questionioning post. From my understanding they seem to advocate a general cognitivist approach to language learning. So they may see language learning as the same type of knowledge as the knowledge in learning subjects such as maths, history etc.

    I thought initially in the CUDDLES the S of story meant a “communicative” approach but it seems it is much more a memory based approach.

    They also seem to favour what is called the interface position, that is they believe that explicit language knowledge can be converted via repetition, memorisation, explicit teaching etc into implicit language knowledge.

    I assume they rely on studies which have shown explicit language instruction as benefitting improvements in learner knowledge, however this does not mean that they show that explicit knowledge has been converted into implicit knowledge since the outcome measurements are based on testing explicit knowledge.

    The first post on CUDDLES and the first CUDDL could be loosely based on research which shows that focusing explicit practice on functional morphology e.g. inflected verbs could help with understanding the meaning of the sentences the inflected verb is used in? However counting letters in words can’t be classed as functional morphology.

    So overall if I understand their worldview on language learning they see it as any other skill to be learned? If so then this is highly debateable.

    It would be good to see what their research base is. I think I shall pop over to post and ask.

    ta
    Mura

  2. 1) What on earth makes you think “Vince Ulam” is a maths teacher?
    2) Isn’t it dishonest to talk about opinion being split on Twitter when there was a remarkable consensus among maths teachers in England and it was almost entirely people who had never taught maths in this country who objected?
    3) “this makes it impossible for students to determine anything by inspection – an important feature of trigonometry” What does this mean? “Inspection” has a particular meaning in maths, that simply doesn’t apply here. You seem to be using it to refer to the act of looking at a diagram and spotting features based on initial appearances. This is not something that should ever be encouraged when dealing with diagrams that are not drawn to scale.
    4) Are you arguing that questions should not get more difficult? If you are not, why would you not expect the potential cognitive load of questions to increase as you go on? (I say “potential” rather than actual here, as greater automaticity may counter-act the increase in the complexity of the question).

    I know you have already seen it, and dismissed its content apparently without taking the time to become familiar with the contents, but if anyone is actually interested in how the angle questions work, I wrote about it here: https://docs.google.com/document/d/1UwmL_o3MDVCLH-0uA6YoSJEKNziNs8k-155mwZ5dGRs No maths teacher seems to have disagreed with any of it, and a lot of the non-maths types who thought there was an issue here changed their mind after reading it or acknowledged there were no major issues.

    • 1. You’re quite right. Corrected to ‘mathematician’ – which is what he (or she) claims to be.

      2. No. The opinions I saw were divided.

      3. In science in general, it means ‘looking at’. I’d be interested to know what the specific meaning is in trigonometry. And in science in general, shape (to scale or not) is an important feature of many kinds of entity. A diagram not being to scale doesn’t justify it being the wrong shape.

      4. No. I’m arguing that cognitive load shouldn’t be increased without good reason. Working memory has the same limits regardless of how easy or difficult the questions are.

      If non-cognitive science types are entitled to make judgements about cognitive science, then presumably non-maths types can do the same about maths.

      • 1) I think that describes the Twitter avatar, not the person behind it. I can’t imagine an actual mathematician not knowing what mathematicians mean by “inspection”.
        2) Not from people who have taught secondary maths in this country they weren’t.
        3) You repeated a comment that you didn’t understand?
        4) Who said anything about “without good reason”? Who are you to judge whether any aspect of these questions was with or without good reason?

        I did not argue with people who were not maths teachers making judgements about maths. My issue was you describing a situation where some people who weren’t secondary maths teachers claimed that all the secondary maths teachers were wrong about secondary maths as showing “opinions were divided”. You might as well say “opinions are divided” about whether the moon landings happened. Or whether “dog” is a word.

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