Experts and the elements of understanding

This is one for the cognitive science nerds. A paper has just been released pre-publication by Ouhao Chen, Slava Kalyuga, John Sweller*. It attempts to unify the well-known ‘expertise reversal effect’ with the ‘element interactivity effect’. I’ll try to put this succinctly but I’ll surely miss some important nuances in the process.

Imagine we are solving a problem and that different parts of the problem rely upon each other. A typical example would be rearranging an equation: We can’t do the steps in isolation because the second step depends upon the first and so on. Similarly, construction of a paragraph will draw upon different elements that each depend upon the other. Contrast this with learning a list of labels. The example Chen et. al. use is learning the Periodic Table in chemistry. It would be a complicated task to complete but learning the symbol for fluorine is not dependent upon learning the symbol for iron so the element interactivity is low (no pun intended).

Van Gog and Sweller have previously used this idea to explain experiments which seem to show that retrieval practice works for low element interactivity learning but studying worked examples is better when interactivity is high. The new paper uses the same idea to reconcile the seemingly contradictory ‘generation’ and ‘worked example’ effects and draws on Chen’s experiments to bring this into focus: For novices, generation works better for low element interactivity and worked examples work better for high element interactivity and for more knowledgeable learners, generation was best all round.

Element interactivity is not straightforward to define. Chen et. al. themselves suggest that processes such as making inferences to construct mental representations, integrating them with prior knowledge, or blocking irrelevant information are hard to define in terms of interacting elements and it is worth pointing out that Van Gog and Sweller’s findings are contentious and a matter of some debate.

If we accept the notion of element interactivity – setting aside concerns that it might sometimes be hard to identify the elements – then it should change as we become more expert at a task. Once we have completed a certain kind of process often enough, that routine becomes like the subroutine of a computer. We can pull it out of long term memory as and when required. We don’t have to deal with all of the interacting elements in working memory and it consumes fewer working memory resources. Element interactivity effectively reduces with increased expertise. This could account for the expertise reversal effect where experts learn more by solving problems than by studying worked examples – the opposite of what we see for novices.

Those who follow the rhetoric of mathematics education reform will also be interested to find that Chen et. al. have a go at defining ‘understanding’:

“Element interactivity also can be used to define “understanding.” Information will be fully understood if all interactive elements can be processed in working memory simultaneously (Sweller et al. 2011). Nevertheless, the term understanding tends to not be used when dealing with low element interactive information. If someone deals with information low in element interactivity, such as “Cu” stands for “copper”, we would not refer to them understanding or failing to understand the relation. If we fail to recall this relation, we will attribute the failure to forgetting or having no prior knowledge rather than failing to understand. Therefore, understanding is only used for materials high in element interactivity.

The distinction between learning by understanding and learning by rote is also related to element interactivity. Learning by understanding increases the number of interactive elements that must be processed in working memory simultaneously. However, if a large number of interactive elements cannot be handled simultaneously, learning by rote reduces the number of interacting elements albeit at the expense of understanding. Of course, learning by understanding is the ultimate goal of instruction.”

Some of this is similar to ideas that I have tried to articulate before about ‘subjective understanding’ – we feel that we understand something when we can manipulate the elements in our working memory. I am a little bit uncomfortable with the second paragraph because I see no reason why we cannot both memorise something and also see how it fits into the bigger picture or ‘understand’ it. So I don’t see that memorisation is necessarily at the expense of understanding. I can memorise a multiplication fact such as 7 x 8 = 56 whilst also understanding why this is the case and how I could demonstrate it to be true should the need arise.

I think that all educators need a basic theory of expertise given that this is what we are aiming at for our students and the idea of element interactivity could add something here. I might also take this opportunity to shamelessly plug my new book (which you can buy here) which addresses the issue of expertise and the misconceptions that we have about it.

*Disclosure: Kalyuga and Sweller are my PhD supervisors


13 thoughts on “Experts and the elements of understanding

  1. Apologies in advance for having so much to say in this comment!
    I, like you, feel somewhat uncomfortable with the second paragraph, so I’d like to try and situate the discussion in a concrete context: the question of teaching phonics for teaching children to read and spell.
    Many phonics programmes do appear to teach children by rote: they put up letters on the board and children say them repeatedly as the teacher points to them. Sometimes, this approach is supplemented by adding in characters (B is ‘Bertie boot’, etc.) or actions as the letters are introduced – extraneous learning, in my opinion! Of course, some children do learn from this and ‘work things out’ for themselves as they go along. However, my contention is that many don’t! This is because the learning of the ‘facts’ (linking letters to sounds) is not explained conceptually (explicitly.) That is, it isn’t made clear that the writing system is a system for encoding the sounds we speak in our language every day are represented by spellings.
    To begin with, over about two to three weeks, we [Sounds-Write] introduce five sound-spelling correspondences [/a/, /i/, /m/, /s/ and /t/] and these are linked to the spellings [, , , , , respectively].
    The way this done is by teaching the skills of segmenting and blending and, by introducing, bearing in mind the limitations on working memory, only three sound-to single-letter spelling correspondences in the first activity/lesson. Thus, low cognitive load – high degree of interactivity!
    So, at this point, what exactly is the child learning and how many cognitive activities are involved? Well, they are learning that, for example, the sounds /s/, /a/ and /t/ are spelled
    , and and that those particular sound symbols ‘stand for those sounds’. They are also learning how to segment in a hands-on activity and how to blend sounds together to form the word ‘sat’. I won’t go into detail about how this exercise is conducted but I’ve explained it here:
    In this lesson, there are three operations going on:
    Facts, in terms of learning that ‘this’ is /s/ and ‘this’ is /a/ and ‘this’ is /t/. As children even as young as three or four can understand symbolic representation – ask a child what this (draw a circle) can be and they sometimes outperform adults – they often give answers like ‘an orange’, ‘a face’, ‘a moon’, ‘a ball’, etc, etc.
    A concept: we have planted the germ of a new idea: letters are symbols for sounds. Of course, the child now has to remember which spelling connects with which sound. This is established relatively easily if the child is given enough practice, i.e. every day.
    Procedural skills of segmenting and blending.
    To summarise, so far, the child has simultaneously been learning that the three spellings stand for three sounds in their own language (English) to form a recognisably English word ‘sat’. (In other words, we have also established a context.) These are facts. They have also (as described in the lesson) earned how to perform two skills: segmenting and blending, through which the ‘facts’ have been introduced. In addition, they have begun to develop the concept that sounds in the language can be represented by the one-letter spellings introduced thus far.
    Is this concept ‘deeply understood’? At this stage, almost certainly not! And here’s where the interactivity of the elements comes in: conceptual understanding will deepen as more ‘facts’ (sound-spelling combinations) are taught always through the same procedural activities (thus keeping extraneous cognitive load to a minimum at this point).
    As things become more complex, understanding deepens further. After learning all the one-letter spellings to one sound, they are introduced to two-letter spellings in the context of something easy (by analogy) to recognise , , and as examples of two-letter spellings but also sensitising children to the idea that sounds can be spelled in more than one way, a concept they will be taught formally in due course.
    I’ve thought a lot about the role of conceptual understanding in the domain of early literacy and, while I think that, as concepts (new ideas) are introduced, from simple to more complex, through the introduction of ‘facts’ and skills/procedures, the whole process begins, even after a short period of time, to run in parallel. I do, though, strongly believe in the importance of continual recycling of the concepts because they provide the structure to which facts and procedural knowledge is bound. And, as we point out on our courses, young children, in particular, often have a tendency to revert to previous thinking unless they are reminded through analogy. They may, from much practice, easily come to grips with the ‘fact’ that, represents the sound /sh/ but will this idea be understood in the context of , , and so on? Some children I have taught quickly begin to generalise these ideas across the domain; others need it to be taught and they need to be reminded again and again. This is especially true of such a complex alphabetic system like English.
    As the richness of the schema develops, the concepts, skills and knowledge of the facts can be generalised all the way across the domain to deal with the greatest complexities: spelling difficulty words, such as ‘diarrhoea’, ‘subdermatoglyphic’, etc, etc. this is, to my mind, an example whereby element interactivity has increased to such an extent that the expertise reversal effect begins to take over. Apart from the concrete experience if seeing pupils’ ability to read and spell ever increasingly complex text, how would we know that understanding has been deepened? Because they can verbalise back to you exactly how it all works and with increasingly nuanced explanations.
    If the rudiments of teaching early literacy are not taught explicitly and to automaticity, it is going to be very difficult to expect pupils to wrestle with the difficulties of dealing with the soon-to-be-faced extra demands of cognitive load: a complex writing system, more abstract vocabulary than is frequently encountered in everyday speech, the greater complexities of the grammatical structures of more academic writing, discourse, and so on.

    • Oh dear, what’s new. When I was at primary school in England (’47 to ’52) we had “The Radiant Way”, with cuh ah tuh spells cat, and only one kid left at 11 unable to read. My second son was a victim of the “Whole sentence” method – disastrous! He didn’t start reading books for fun until he was 17 years old. And so the wheel turns.

  2. I think that the division into “learning through understanding” and “learning by rote” is too simple minded, and misses a whole lot of stuff learned, in the sense of “firmly located in long term memory”, by repeated exposure, familiarity, repeated usage and similar.
    Example 1: “Twas slithy in the borrowgroves”. I sure don’t understand it !! Thankyou, Edward Lear, but I didn’t “rote learn” it.
    Example 2: 7×9=63. I learned that 8×8 is 64, by rote, I learned through understanding that (n+1)x(n-1)=nsquared-1. I saw that 7×9=(8-1)X(8+1), so magically 7×9=63
    Example 3: In primary school we were told about the longest word in English. Did I learn it by rote or did I analyse the sounds and learn the sound sequence by rote, or did i understand the meaning of the word from its component parts? I don’t know.The word is antidisestablishmentarianism, and this is the first time I have ever written it out!!!!!

    • Brian says:


      I believe you learned the word “antidisestablishmentarianism” by simple chunking.

      You suggested that you shared the concerns of Greg regarding paragraph 2. Greg said ” I see no reason why we cannot both memorise something and also see how it fits into the bigger picture or ‘understand’ it.”

      To quote Dan Willingham from memory. Understanding is remembering in disguise.

      I believe you must know something before you can understand it. You can memorise information in different ways. You can remember a fact without understanding it. You can memorise how a fact fits within a bigger picture without understanding it. To understand it you must know why it fits into the bigger picture where it does.

      When you understand it you will memorise by organisation. When you remember it you memorise it usually by repetition.

      This paper appears to confirm that which is already known. This paper will make it difficult for people who do not like to draw a distinction between knowing by rote and understanding. Greg fits this description I think. As soon as Greg accepts this one he will be moving over to the light side.

      The idea of element interactivity can in my view be represented well using concept mapping, a concept that Greg has recently dismissed as a waste of time. There is hope for the poor chap yet.

      • Chester Draws says:

        The idea of element interactivity can in my view be represented well using concept mapping

        I want to know how to teach my students the interactivity of parabolas and quadratic equations. I do it every year, and it’s an issue.

        Show me how that might be represented well by a concept map.

        The problem is that the concept map shows that they are linked, but doesn’t show how they are linked.

        Concept mapping suffers from this fatal flaw. It asserts a link, but being diagrammatic can never actually explain that link in any deep way. It remains a method useful only for showing surface features in common.

        This paper will make it difficult for people who do not like to draw a distinction between knowing by rote and understanding

        I’m lost how this arises from the paper.

        We can’t ever “know by rote” anyway. We memorise by rote so that we have something in our memory available for instant retrieval. That’s it. The understanding is separate from the memorising, but made much easier because the facts are at instant recall so we don’t waste cognitive resources on them.

    • I agree that the division is too simplistic. There are at least 3 categories:
      1 learned by rote and not understood
      2 understood, but not secured in long term memory
      3 understood and secured by repeated use/practice.

      The process of moving from 2 to 3 may look like rote learning, but is quite different.

  3. stan says:

    I am intrigued how you discuss the relationship between a parabola and quadratic equation. Here in Ontario Canada the issue is avoided by a circular argument. See

    Click to access math910curr.pdf

    There a parabola is defined as the graph of a quadratic in the glossary while in the course description students are asked to “determine,through investigation using technology,that a quadratic relation of the form y = ax2 + bx + c (a ≠ 0) can be graphically represented as a parabola.”

    I find that pretty funny for a 21st century curriculum that aims to have students thinking like mathematicians (you know those people that really care about definitions and what it means to determine something.)

    Fortunately the Wikipedia entry offers a nice derivation from the directrix and focal point definition of a parabola to a quadratic equation. Going the other way seems quite a bit less intuitive.

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