Why should we study complexity? Reflection 1 on #Touchpaper problem party

For a context for this reflection, please read the previous two posts.

In this blog I have compiled a short list of the reasons why determining the complexity of a concept is so important for teachers. In subsequent blog posts I will unpick some of the discussions that we had when trying to solve this knotty, philosophical question.

Determining the complexity of a concept is important because…

1) … it makes you think about the order you do things in as a teacher.

In our discussions on Saturday we identified three broad stages each teacher needs to go through to help their students understand difficult concepts:

a) Map out their topic area into linked and ordered concepts.

b) Consider what barriers and difficulties students might experience in trying to understand a given concept…outside of the inherent difficulty of the concept.

c) Check for their students’ conceptual understanding…preempting and engaging with misconceptions.

2) … it helps you pitch work at the right level for your students.

If you work out how complicated a concept is, then you can work out how quickly your students might grasp it, what prior stage in their understanding they might need filling in, what they might need reminding of or, alternatively, what they might already find too easy.

3) … you can become more confident in sequencing units of work.

How do you order the concepts in your subject? Historically (as they were discovered), chronologically (as they ‘happened’), empirically (by working out what most people know vs what most people don’t know), in order of complexity (simple to complex) or spiralling (to repeat the most difficult of concepts several time)? Or a mixture of some/all of them?

4) … it identifies gaps in your own subject knowledge.

I remember being told at teacher training that if you can’t teach something to someone else, you probably don’t have mastery of it. Now while this might not be true in the completely literal sense, there is a grain of truth behind it at least: if you cannot break down complexity in your subject area in order to explain it to someone else, you probably need to do some more work getting to know the subject area! Mapping out your subject concepts is very useful for identifying these areas.

5) … it provides you with a diagnostic for correctly identifying your students’ misconceptions.

If a student doesn’t ‘get’ something, you can use your understanding of how the concepts within your subject area link together to identify dependent concepts that need to be taught prior to the student being able to ‘get’ the thing you were trying to teach them in the first place.

6) … it can help you pre-empt difficulties.

Mapping out complexity and considering why students might find something difficult means that you may be more likely to choose a more appropriate range of teaching techniques to suit the teaching of that topic. When you stop to ask yourself if a concept is counter-intuitive, or if it requires a lot of prior knowledge, or whether you have prepared sufficient examples to explain it, then you are more likely to have identified a difficulty that you can plan for.

7) … it involves identifying threshold concepts that probably take more time to teach.

Subject disciplines have paradigms particular to themselves. Some of these are counter-intuitive, others can be considered ‘troublesome’ because in understanding them, you subvert a preconceived notion or previously held knowledge. The idea that light can be considered a wave or a particle in physics. The idea that a reader constructs meaning in literature. The idea that cooking is all about energy transfer. These have been termed threshold concepts and may need more time devoted to them as they are so crucial. Once you grasp a threshold concept, it’s very difficult to forget; it shapes your understanding of the whole discipline. In future blogs I will look at some examples to see how this applies across the subject disciplines.

8) … it could help you work out an assessment structure.

In order to write an assignment with a metric of some sort, teachers encounter the need to devise a mark scheme or set of success criteria. Within each subject these are different depending on the stage that the pupil is learning at. This forces the teacher to determine the complexity of a concept because they have to find out at what point they can award a top mark, at what point they differentiate for complexity and how much of the complexity the students need to grasp at the given stage in their learning. By way of example: a top mark for an 11 year old’s essay about a piece of literature, would be less difficult objectively speaking than the top mark for an undergraduate analysing the same text.

These are some of the justifications for why this #touchpaper problem is worth deliberating further.


Party Planning: My thoughts on #Touchpaper problem 4 – what determines the complexity of a concept?

On Saturday I will be facilitating the #Touchpaper Party organised by Dr Becky Allen and Laura McInerney at the IOE. Here are some brief thoughts on the problem that my group will be tackling. Read a background to this here.

Problem #4: What determines the complexity of a concept?

Laura McInerney has put this problem into a teacher’s context for us here. If you haven’t already, please read it first. For most references to Wittgenstein I am deeply grateful to Dr John Taylor for sending me his excellent lectures on the philosopher which distilled some very complex concepts indeed!

Proposed definitions:

  • Complex: made up of more than one. (Not to be confused with difficult).
  • Concept: an essence; an idea; a process; a category of real or imagined existence.
  • Difficult: finding something not easy, often requiring more knowledge resulting from a cognitive overload.

Here are some “givens” – both premises, observations and assumptions.

1. It’s personal, and therefore relative.

“There is a temptation for me to say that only my own experience is real: ‘I know that see, hear, feel pains, etc., but not that anyone else does. I can’t know this, because I am I and they are they.” (Wittgenstein, Blue and Brown Books, p. 109)

We are all so complex ourselves that there are several reasons why we might understand things in different ways: our brains are wired differently, we perceive things differently, we make different assumptions and so on.

However, in theory, the teacher has control over one variable which unlocks this dilemma: teaching a whole group the same thing to bring them to the same level of knowledge and understanding.

Except in practice, that doesn’t always happen because pupils’ prior knowledge is so varied and mastered to such different degrees. And knowledge cannot be abstracted and isolated from context.

This is why difficulty of understanding is relative: because what might be considered easy for one pupil is difficult for another.

Equally, the complexity of any given concept is relative: as Laura suggested, a 7 year old and an 18 year old can all understand ‘appeasement’, but to different levels.

2. Context is crucial to understanding.

“If the development of vocabulary knowledge substantially facilitates reading comprehension, and if reading itself is a major mechanism leading to vocabulary growth which in turn will enable more efficient reading-then we truly have a reciprocal relationship that should continue to drive further growth in reading throughout a person’s development.” (Keith Stanovich, Matthew effects in reading: Some consequences of individual differences in the acquisition of literacy. 1986)

If the concept is presented in a context where little related fact is known, it becomes more difficult. If it is presented in a context which is familiar it becomes easier.

The more you know, the easier it is to learn more, as Stanovich demonstrated in his seminal paper on the Matthew Effect in reading. 

The analogy of the Matthew Effect can be applied to our acquisition of knowledge through a range of metaphors. It could be said that knowledge is a net in your mind whose mesh becomes finer and finer the more you know, and makes it easier to trap and hold on to new knowledge that falls through your mind. It could also be said that knowledge has mass – the more you know, the larger the mass and the larger the gravitational pull, making it easier to attract new knowledge and ensuring it attach itself.

Context can also change meaning, however. Depending on how the concept is used, it will mean different things.

3. Language is crucial to understanding.

“It seems that there are certain definite mental processes bound up with the working of language, processes through which alone language can function. I mean the processes of understanding and meaning.” (Wittgenstein, Blue and Brown Books, p. 3)

 So if we delve into each student’s mind to work out what they know, how they understand and at what level of complexity they engage with a concept, we bump up against the transmission of knowledge through language. 

Language is tied up with meaning and understanding. And conceptualisation.

This leads us to suggest that the conduit metaphor for language might have its limitations. Could it be said that the language is the knowledge or this is an oversimplification?

4. Clarity of language is clarity of thought.

“[The English language] becomes ugly and inaccurate because our thoughts are foolish, but the slovenliness of our language makes it easier for us to have foolish thoughts. The point is that the process is reversible. Modern English, especially written English, is full of bad habits which spread by imitation and which can be avoided if one is willing to take the necessary trouble. If one gets rid of these habits one can think more clearly.” (George Orwell, Politics and the English Language)

Whether it is an oversimplification or not, the language that someone uses provides us with plenty of clues as to how well they understand something. The more clearly they write or speak, the more clearly they probably understand the concept.

Here is a suggested flow diagram of the expression you make of conceptual understanding:

A. what a concept means in a given context > B. what you understand the concept to mean > C. what you intend to say > D. the language you use > E. what is understood.

Let’s assume that A is objectively true. At every other stage there is scope for misunderstanding and a lack of clarity: you could be guilty of mis-conceptualising in the first place, you could be guilty of meaning something illogical or untrue, you could be guilty of choosing the wrong language to express yourself, you could, in turn, be misunderstood.

5. Language can be a barrier.

“Texts in social sciences and humanities tend to be loaded with grammatical metaphor, as they construct and evaluate abstract concepts about social life.” (David Rose and J.R. Martin, Learning to Write, Reading to Learn, p. 195)

Language can obfuscate (as George Orwell suggests above) or it can just be difficult for a student, especially if the concept is complex or abstract enough so as to be beyond their current level of knowledge.

Therefore we need to unpack the complexity of the concept. This often means ‘de-nominalising’ abstractions – “by turning them back into activities that involve people and concrete things, and making logical relations between activities explicit with conjunctions” (Rose and Martin, p. 196).

Denominalising means unpacking the noun form. For an abstract concept, a noun is a proxy for the abstraction – a noun is not a name or an object.

Not all nouns are names. Concepts are often nominalised (i.e. turned into nouns). In order to unpack them we need to denominalise them. Grammar can be misleading here. If I ask “what is the sun?” that is not the same as asking “what is time?” The sun is an object. Time is something more problematic, and intangible in comparison.

Metaphor can also be misleading, especially when it is embedded in what we think of as ‘correct’ language.

But not all concepts are abstract concepts.

6. Concepts that can be understood scientifically are actually processes and can be sequenced in order of complexity.

“You can’t understand electrolysis unless you know what an aqueous solution is.” Juan Casasbuenas, in the staff kitchen, yesterday.

First you explain what an ion is, then solutions, then electrolysis…

There is a sequence to teaching concepts that are actually processes. The best sequence is subject-specific and determined by the learner’s prior knowledge.

The teacher: identifies the end-point concept that must be understood, ‘reverse engineers’ the concept to its constituent parts, sequences the constituent parts into the most logical order drawing on their own subject expertise to do so, identifies at which point along the sequenced conceptual continuum the learner currently finds him or herself and begins teaching from that point. This is scaffolding and sequencing and planning as we know it.

The learner: relates to the material that is first being taught because they recognise it. They then build new knowledge onto this, constructing their own understanding with the help of the teacher. 

We can also see that for certain complex concepts it is possible to break them down into their constituent parts. This would be a scientific approach. 

7. We need different ways of thinking: scientific and philosophical.

“Wittgenstein claimed that this obsession with generality was a result of our obsession with science. Science seeks to reduce the variety of the natural world to a few, primitive natural laws. It seeks to explain the world’s diversity in terms of laws which are framed using natural kinds of property, such as mass, force, temperature, charge, etc. The scientific method is tremendously powerful, and it is tempting for philosophers to emulate it – to try to reduce the diversity of language to simple, generally applicable definitions. But this tendancy to copy science, in Wittgenstein’s opinion, leads into ‘complete darkness’.” Dr John Taylor, lecture on Wittgenstein.

What Wittgenstein refers to as generality is the tendency to believe that a concept can have a single, simple, definition. This goes against what I have suggested in point 2.

This summary of Wittgenstein’s thought in the quotation above demonstrates why a philosophical approach is necessary to answering this question; when it comes to understanding and meaning (both complex concepts in their own way), we risk running around in circles if we’re looking for generality, when meaning can change depending on the context of the concept.

That said, there is clearly a scientific approach that is relevant in certain contexts.

So what next?

Suggestion for how to tackle Touchpaper problem #4.

Behind the problem as it is currently phrased are the following ‘sub questions’ which can be addressed in turn:

1) What is ‘meaning’?

This will address some of the problematic philosophical questions that arise and will therefore require a philosophical approach, drawing on subject specialisms. Some suggested topics could include:

    • how language is used to create meaning
    • how the essence of an idea remains separate from the language that we use for it

2) How can I tell if my students really understand?

This will address the issue of understanding being personal to each student and of ascertaining the prior knowledge of each student, as well as working out how much they understand along the way to conceptual mastery. Some lines of inquiry could include:

    • how does what students say and what students write help us gauge their understanding on a scale from novice > expert?
    • what range of techniques is required in order for a true assessment of learning to take place and which techniques assess for which type of learning?

3) How does one best scaffold and sequence content for mastery?

This will address the issues of building complexity, especially for those concepts which are also processes. Suggested approaches could include the following:

    • What is the best way to scaffold the introduction of new knowledge for linear or scientific complexity (that deals with cause>effect)? This would have to be addressed by subject experts, perhaps with reference to threshold concepts.
    • Is it possible to teach conceptual mastery through teaching language and grammar? This could draw on the advances of genre theory (a defence of this approach by Frances Christie, addressed to Michael Rosen, complete with reading list) and language-based pedagogy (a link to what this looks like in practice, with thanks to Lee Donaghy).

Please do have a read and let me know what you think, folks.

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

Laura McInerney

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…

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