TouchPaper Problem #4 – What determines the complexity of a concept?

This is the fourth blogpost expanding on the TouchPaper Problems first discussed at #Researched2013 and due to be tackled at the first TouchPaper Problem Party.

Question #4 – What determines the complexity of a concept?

In my estimation, this is the hardest of all the problems, but it’s also really important.

As a teacher I was constantly trying to figure out “how difficult is this material?” and gauging whether I needed to edge it up or scale it down depending on the students I was teaching. But: how do we know if something is complex?

I remember working with a history revision class who were learning about the appeasement. I didn’t think appeasement would be a difficult concept. I mean, kids appease each other all the time in the playground when they, say, allow an older bullying set of kids to play football with them, even if they don’t really want to, in order to avoid a conflict. Problem is, appeasement actually turned out to be quite complex. “To appease” is easy to understand. “Appeasement”, however, is a strategy, a non-concrete object, and it’s quite difficult to talk about accurately without practice. My students kept saying things like “Britain wanted to appeasement Hitler” or “the appeasement happened in Munich” – and while both are in a ballpark I could understand, they were inaccurate enough that they couldn’t go uncorrected.

A second thing led me to create this problem. Earlier the year national curriculum levels were “abolished” and schools are now being encouraged to create their own. Many vocal opponents of levels complained that the stages did not adequately follow on from one another, with some actions described at Level 8 not necessarily seeming more difficult than those at Level 7. Others suggested that what we should have instead are lists of knowledge that students will have and that this should get progressively more difficult as we go forward.

But: how do we know which knowledge is the most difficult? To go back to the issue of appeasement, I’m fairly certain I could get a 7 year old to understand much of it. There also people who write their PhDs about it. So what is the essential difference between the types of concept the 7 year old and the PhD are using when discussing appeasement?

As with the other problems, I am certain there has been lots of study on this. Taxonomies of knowledge exist. Philosophers of knowledge have hierarchied such things on occasion. But what I want to know is how these tools can help a teacher know the complexity of a concept. Because if we can answer that question then we can start to construct assessments and curriculum on the basis of some collective understanding.

Getting to that point, however, seems like will be far from easy. Any suggestions?

Categories: #TouchPaper Problems

40 replies

  1. Reblogged this on Mr Lock’s Weblog and commented:
    I am fortunate enough to be attending the inaugural TouchPaper Problems party on 19th January. This is the question I have been allocated to work on all day. As Laura says, it’s a really difficult question. I should probably get on with some reading in advance of the 19th – which I’m looking forward to.

  2. Reblogged this on Ready for the New World and commented:
    I will be attending the Touchpaper Problem Party in London in a couple of weeks, and will be working on this problem. Certainly, with all my years in the special education classroom, I have given this a good deal of thought, but it’s difficult to put them all into an organized structure practical for planning instruction. Any thoughts out there that I might bring with me?

  3. Hello! I am very excited to be facilitating this session and really look forward to hearing everyone else’s thoughts on this issue.

    To echo Laura’s introduction to the question, this is a very difficult problem to tackle. Much thinking about this seems to go in circles as words such as ‘knowledge’, ‘concept’ and ‘complex’ are nebulous insofar as they can take on very different meanings depending on how they are defined. Moreover, this Problem links to other phrases we all come across and use as teachers such as ‘higher order thinking’, ‘open vs closed questioning’ and ‘scaffolding’.

    My efforts as facilitator in the build-up to our party (whoop!) will be to define the terms of the discussion, which is no simple task in itself. Any suggestions made here will be happily considered, in particular from anyone who has a grounding in philosophy or logic! I hope that once a working definition of these key terms is agreed upon, then the question can be broken down into more manageable lines of inquiry.

    To share my thinking so far:

    My initial attempt at addressing this question was to ask ‘what is one unit of knowledge?’ This is essentially a restating of the initial problem – ‘how do you determine the complexity of a concept?’ However, exploring this problem forced me to start to work out whether we should bother defining complexity in the first place. This might seem counter-intuitive, but I suspect that this is a fruitless task. After all, knowledge doesn’t exist in some kind of series of silos or honeycombs but forms part of a network with other knowledge. The concept of ‘zero’ varies in complexity whether it is understood as part of a number line (…, -2, -1, 0, 1, 2, 3, …); or whether it is understood as a numerical representation of a lack of ‘tens’ in a base 10 counting system (as in the number 101 for example); or whether it is understood as a historical revolution in mathematical thinking that proved to be the gateway to the concept of infinity and the concept of negative numbers. Given that any one bit of knowledge, let’s call it a ‘fact’, differs in complexity depending on the context in which it is situated, we risk a pointless, circular quest for some sort of ‘Grand Unified Theory’ of knowledge (I borrow the term from physics where the GUT posits that there is a simple number of constituents that make up all matter. Apologies for getting the wrong end of any stick here! –

    So what happens if we don’t bother defining complexity? Well then our role is very much to work out how complexity is best *structured* for understanding. As a qualified teacher, I like to think that I’m quite good at planning stuff, so in the best, time-honoured tradition of ‘planning backwards’, it might make sense for us to be able to ‘reverse engineer’ the outcome we want to see in our students (complex, conceptual understanding) without going so far as to suggest that understanding can ever *not* be complex; maybe understanding, be definition, demands complexity?

    If my reasoning up to this point is correct, here are two suggested directions that our group might perhaps want to go down. I do not see them as mutually exclusive:

    1) Using taxonomies of knowledge. Explore what research exists that ranks complexity of understanding, focusing on what outcomes we would expect of our pupils at each given stage. This serves the dual purpose of making the abstract concrete, while also narrowing our focus to something achievable in the time available.
    As a starting point I’d like to suggest this article on revising Bloom’s Taxonomy of knowledge by David Krathwohl (not because it is necessarily definitive, but, rather, because in revising Bloom’s original taxonomy, I discusses some of the issues touched upon here):

    2) Linguistic approach. We first define ‘complex conceptual understanding’ as using words in different word classes (e.g. if a pupil has complex conceptual understanding of ‘appeasement’, then they correctly *use* ‘to appease’, ‘appeasement’, ‘appeasing’ – the verb, the noun, the gerund). Taking this as our outcome, we should therefore work backwards to see what research exists that teaches concepts with an emphasis on linguistic and grammatical approaches.
    – A lack of conceptual clarity comes across when pupils speak or write. In particular, it might therefore be of interest to look at what theory and advice already exists that helps teachers most improve pupils’ speaking and writing.
    – It might be helpful to look at how teachers address ‘misconceptions’. The work that Daisy Christodoulou helpfully collates in her blog ( might be a good starting point for this.

    Anyway, these are just some initial thoughts. Maybe I should do a proper blogpost prior to the day. However, please do send me any links, thoughts, arguments and questions in the meanwhile!

  4. This is a tough one.

    How about: “The complexity of a concept is proportional to the number and scope of arguments that need to be deployed to support or justify that concept.”

    For a Year 7 student, the concept of appeasement can justified using a playground analogy and comparing the actions of governments during the interwar period, perhaps. A PhD student would marshal many more arguments of greater depth and greater scope…

    Just a thought.

    • I like this because it suggests another way of ‘ranking’ complexity in a relatively simple way. I am minded of this article by Laura explaining how she uses SOLO taxonomy in this regard – it can be a useful diagnostic of the student’s output and helps us categorise a level of understanding at which students are at:

      Again, this suggests you would agree that the best way of tackling this problem is to look at student outcomes first, and work backwords – i.e. if we want to increase complexity of conceptual understanding it is a matter of getting the students to increase the number and scope of arguments that they are using. Now it’s a matter of working out how you can most efficiently increase the number and scope of their arguments…

  5. This reminds me of the way that ‘atoms’ or acid/bases was taught to me in Chemistry over a period of 6 or so years. It is clear that the most accurate description of an atom (basically high level maths) would be incomprehensible to school aged students. So, instead, the various simplified models of the atom (starting from the Greek indivisible building blocks of substances) were taught (which also happens to be the chronology of discovery, too) first. Each year would start with the teacher saying “You know what we taught you last year about the atom? Well……”, and out would come the next, more complicated and more accurate model of the atom. Over time this taught us that in Science knowledge evolved; and that the knowledge being taught had probably been adjusted to fit our capacity to understand. It is axiomatic in Chemistry that all knowledge and understanding is dependent on prior knowledge and understanding.

    Complexity, in this case, depends on what you already know; the pathway the individual has taken to arrive at the learning opportunity….and the realisation that where you currently are at isn’t the end of the story.

  6. I need a little clarification in my understanding of the problem. Are we discussing the levels of complexity of the student’s understanding of the concept, or are we discussing the complexity of the concept itself? That is, are we agreeing beforehand that all concepts have these continuous levels of complexity and we are discussing ways to effectively use our (by this time, well-honed) formative assessment skills to determine where students are in their understanding and how to move them along the continuum of greater understanding, or are we looking at individual concepts and saying,”How complex is this one?”

  7. My favourite topic so I will watch with interest the way the comments develop.

    I find “complexity” to be an interesting concept in itself. When a property of the concept of “concept”, even more fascinating.

    As a constructionist at heart, I believe concepts and the level of complexity at which they are addressed is the key to effective teaching and learning.

  8. ‘What determines the complexity of a concept?’
    In an educational context, how difficult it is to bring it to life in someone else’s mind – to get them to understand what it really *means*. If a teacher is able to make a concept concrete, it’s an easier concept, a ‘ground level’ one (e.g. dramatic tension, symbolism). If I have to use more abstract ways to show you, to get you to see what I mean, then it’s a harder concept (e.g. metaphor, irony). My job is to find the ways to show you that will make the most sense to you. To help you weave them into your current level of understanding.

  9. What determines the complexity of a concept?
    Possibly the breath and depth of the schema involved in cognitive, emotive and social terms.
    This may also depend upon other related concepts as well as what is known, not known and unsure.

    Whether the concept involves forms of necessary hierarchical understanding, i.e. things one needs to know before one can know something else might also add complexity.
    Other complexity might arise from inherent possibilities of misconceptions that arise around the concept as well as barriers that might exist in gaining such knowledge, physical or social for instance.

  10. I would suggest that complexity of a concept is relative – several other comments here are making this point – and that it is a question of the number of steps between a person’s current knowledge and understanding and the knowledge and understanding needed for the concept in question. Thinking about the role of assessment in curriculum design, I suspect what we do as teachers, is look at the assessment requirement for a concept, and then consider what we need to do to get our students from where they are now, to being able to be successful in the assessment. How many steps there are is how complex the concept is, and dictates how long it takes to teach. If ‘appeasement’ requires three lessons to get students to be able to write a credit-worthy answer to a GCSE question on it, and ‘reparations’ can be taught in one lesson, then for those students and that teacher, ‘appeasement’ is more complex than ‘reparations’. In fact it is three times as complex – ooh I’ve gone all quantitative. This argument suggests that the question is about how to define one piece of knowledge, and then how to add up all these pieces (including practising) to find the total number of steps. I find myself doubting that this could ever be done accurately enough to beat an experienced teacher going ‘hmmm – should take three lessons’. Maybe the end-point of this question is to ask how teachers could better collaborate in case there are a load out there that make a great job of teaching ‘appeasement’ in two lessons whilst the three-lesson teachers are under the illusion that this concept is 1.5 times more complex than it really is. Good luck on the 18th.

    • Hi Dodiscimus, thank you for a very interesting few points here. In particular I like the way you tie the discussion to the question of assessment. This is perhaps a better way of phrasing what I also meant when talking about the ‘outcome we want to see in our students’ earlier in the thread.

      In drawing up a set of success criteria – a mark scheme, say – how do you end up ranking the mark descriptors? Do you begin by sorting out what is less complex from what is more complex?

  11. The number of steps/stages required to explain it?
    The complexity of the ideas that make up the concept?
    The proprtion of abstract vs. concrete components of concept?
    The reading age or the complexity of the terms that comprise said concept ?
    Just a few ideas off the top of my head…

  12. Since we are addressing this problem as something of education’s version of Hilbert’s 11 problems for mathematicians, maybe it would be good to establish our “givens” before we get started on the 18th or, if it is more a problem of logic, we could think of them as the “a priori” concepts that we are all in agreement about before we get started on the 18th. That way, we can really dig into the meat of what is new and undecided about this topic. Here is one idea, for example:

    Given: For any concept, its complexity for any given students is dependent on the student’s prior knowledge and measured understanding of the concept.

    What would be some other givens?

  13. I think part of the problem with judging “complexity” of a concept is that it is, to some extent, a personal thing.

    It is quite possible that two students, with equal starting points, would find coming to terms with a new concept/idea very different. For one the new area may be relatively straightforward (ie. suggesting that it is not that complex) for the other it may be very challenging (ie. suggestion of high complexity).

    So we need to clarify what we mean by something being complex – is it one which a high proportion of individuals judge as being complex; one with a high mean level of complexity?

    I’d argue that for a concept to be judged complex it needs to meet both of these criteria.

    • Interesting – why would these two students experience this difference? There are some hefty questions for cognitive psychologists here but to my mind, if one finds it easier, then although they may have been at the same starting point with regard to this topic, something in their prior learning makes the difference e.g. one has lots of experience with abstract concepts and one doesn’t. However, I think you are absolutely right to think that complexity is not the same for each individual and thinking about how to establish some sort of averaging process is necessary. I go back to my earlier point, though – can you do any better than getting an experienced teacher to offer an expert opinion. Maybe averaging expert opinions?

  14. So first off, do we need to be comfortable with a definition of ‘concept’? (if you don’t like wiki then go to the references at the bottom of the page for the sources).

  15. I think that looking at the complexity of the Knowledge/Facts themselves is barking up the wrong tree. I’d say even very young kids can grasp complex problems – see this talk for example:

    Could there be a complexity gradient relating to the child’s ability to Contextualise the knowledge? And maybe the teacher can gradient their teaching content according this ability to contextualise the facts learnt. Ie:

    – First the child learns how the facts apply to themselves and their life specifically (appeasement and playground example)
    – then the child can apply this simple concept to their life/society in general (wider examples about how confilcts can be avoided)
    – Then the child can learn/develop techniques to apply the facts to new situations
    – etc.



    • Hi Dan, this is very interesting. Would you agree that this builds on the suggestion of Steve and Dodiscimus earlier in the thread that complexity of understanding is a very personal matter? Depending on a student’s experience, they might be able to access something more complex, whereas someone with no prior knowledge or relevant life experience may well seem to find the same concept more difficult.

      I would be interested in other peoples’ thoughts on the gradient of complexity that you suggest as a result. Would it work as a practical tool to structure the teaching of increasingly complex concepts?

  16. I think this is an extremely interesting problem. I would endorse the second approach suggested by Michael – that it is tackled by engaging in some linguistic analysis. I think, quite generally, that there is an element of the teacher’s task which overlaps with that of the philosopher’s – and it is exactly when we ask questions about the nature of concepts and whether they can be arranged in a hierarchy of complexity or difficulty that the overlap becomes obvious. The most instructive general account of the relevant philosophical background was provided by Wittgenstein, in his later work in particular. For the later Wittgenstein, acquiring mastery of a concept is akin to learning to play a game – it is a matter of gaining a practical mastery of the rules which govern the use of a term. The analogy between games and language is helpful, suggesting as it does that language use should be seen as a social activity, and thus that we cannot understand the meaning of a word in isolation, but need to appreciate its significance as an element within a human social practice. For Wittgenstein, philosophical puzzlement arises when we fail to appreciate the way in which our own language really functions. We are deceived by language itself, on account of the fact that the ‘surface’ grammar of a term might not match up with its ‘depth’ grammar. This observation could explain the way in which Laura’ students misappropriated the concept of ‘appeasement’. The use of an abstract term lends us to think that there is ‘something’ being named (Wittgenstein traces the theory that all words are names right back to Augustine). So it is natural to form sentences referring to ‘the appeasement’ and ascribing it properties which belong to concrete particulars, such as a location in space. Wittgenstein would propose that the remedy for this misconception is to attend to the verbal forms from which the concept of ‘appeasement’ has been abstracted – to consider what it means for one person to appease another, or for a nation to adopt a policy of trying to appease another nation. By replacing the abstract noun with verbal phrases, we can begin to clarify what is really meant by talk of appeasement; and we see too that there may be quite different meanings, since the word may figure in very different language games (placating a playground bully may be to some extent similar to the game played by international statesmen on the brink of conflict, but there is a great deal of political content to the latter language game, which has to be grasped if ‘appeasement’ in its political sense is to be understood.) Apologies for the homiletic tone of this post – if anyone is interested in doing some background reading on Wittgenstein, I’d be happy to suggest starting points..

    • Hi John, I would love to tackle some of the background reading that you might be able to suggest. Perhaps if you reply on here then others could give it a go also?

      I really like the analogy of learning a concept to playing a game. One’s aptitude at mastering the rules of a game, in order to be able to play it better mirrors the idea that a mastery of a concept is demonstrated by a student being able to refer to and use that concept better – something that we measure linguistically.

      There is another aspect to the analogy between learning a concept and learning a game, which is that once you become very good at a game, you can learn to cheat. Could that be analogous to learning to subvert, satirise, de-construct and/or question a concept?

      Like I said earlier, and in agreement with Becca Leech’s latest contribution earlier in the thread, I think that before we come together to tackle the problem, it would be good to give some thought to the definition of the key terms we will end up discussing and the ‘givens’ that we can base any further discussion upon. I suspect that in some instances it will be a matter of just ‘drawing a line in the sand’ – not quite arbitrarily, but recognising that the semantic debate can continue endlessly and it would distract from the task of looking at the more practical solutions to the problem illustrated by Laura in the post that current research might suggest for us in the classroom.

      Thanks for your contribution to the discussion!


  17. As long as the linguistic analysis doesn’t lead us to dancing on heads of pins… 🙂

  18. I think an appropriate way to establish how complex a concept is would be to analyse the network of dependencies the concept has. The more dependencies, the more complex the concept is as it relies on a greater number of previous concepts.

    We must be careful not to conflate complexity with complication (i.e difficulty in understanding). They are different things: A student might find a simple concept complicated (difficult) where as his peer finds a more complex concept uncomplicated (easy). I expect how complicated a student finds a complex concept will likely depend on how clear their understanding is of the dependencies.

  19. It’s clever that this page has now become an example of increasing complexity. I understood the first post. By Wittgenstein I was befuddled. Looking forward to thinking more on Saturday: making the complex simple.

  20. Hello all – better late than never – have put a few thoughts down on this here:

    Looking forward to seeing some of you tomorrow. Any requests for snacks?


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