On Difficulty

What makes something – a subject, a topic,an idea, an ability, anything – difficult?

I found it easier to think about difficulty in relation to abilities, things we do. Is a 360° backflip on a BMX bike difficult? Well, I would find it so. I can’t speak for you. But I can probably speak for somebody who had already done a 720° backflip. They would probably say “No, a 360° is not so difficult.”

So, the difficulty of a particular ability is at least partly in the mind of the practitioner. And that difficulty is great or small at least partly in relation to what they already know they can do.

Staying with ability, and shifting from an individual to a societal view, another indicator of difficulty may be the smallness of the proportion of the population that can do it. The rarer, perhaps, the harder.

Staying with ability – we’ll get onto difficulty in relation to subjects, topics and ideas, l promise – other popular or intuitive dimensions of difficulty may be complexity and scale. I can play any particular chord from Tchaikovsky’s first piano concerto – give me a minute – but not all three movements. I can climb onto a small rock, but not climb Everest.

So, as elements or indicators of difficulty in practical tasks, I can offer:

1 Relation to my current capabilities: “It’s hard because I can’t do it (yet)”, which is “I haven’t done it yet” with the possible addition of “and I can’t immediately see how I would do it.”

2 The proportion of the population that can do it: “It’s difficult because lots of people can’t do it.” (Of course there are lots of things that most people don’t do mainly because they would never dream of or want to do them, but we’ll ignore these for now.)

Implicit in both the BMX and the Tchaikovsky examples:

3 The time taken to learn the ability: “It’s hard because it will obviously take a long time to learn to do it” and

4 The effort taken to learn the ability: “… a lot of work … “.

⁃ And then:

5 The complexity of the ability, the number of different elements and relationships in play; “Good grief but that looks complicated” and, related but not the same,

6 The scale – which may be size, duration, whatever applies. “Gosh but that looks enormous.” The Tchaikovsky example illustrates this well.

These are six possible dimensions of difficulty of abilities. What others do you see?

OK, at this point in the writing process, I admit I am surprised. If you’d asked me before I started what I was going to conclude, I’d have predicted that I would just say “Difficulty is purely, or mainly – i’m not sure which – subjective.” I enjoy writing when it gets me to an unexpected conclusion. The unexpected conclusion I’ve got to so far is that it is possible to talk about more-or-less objective factors which may contribute to difficulty.

But there is a sense in which my initial reaction was right, although of course badly incomplete. Difficulty is a feeling. Time or effort or complexity or rarity, any of the six dimensions of difficulty teased out above, may or may not bring with them a feeling of difficulty. For example, to the persistent or the individualistic, the large time investment required or the fact that the vast majority of people can’t do it may not be problems, may not be sources of difficulty. They may simply be facts. Or they may even attract, appeal.

As promised; what makes a subject, a topic, an idea, difficult?

I think, all of the above, as explored briefly here:

1 “I don’t understand it” and, perhaps, for a more persistent learner, “I can’t immediately see how I would come to understand it.”

2 Maybe “Very few people understand that.” or “Everybody says it’s really hard to understand”, a norm-referenced or crowd-sourced account of difficulty.

3 and 4, perceived or actual time and effort required, are very similar for ideas and for abilities. In education, time and effort may be reflected in the length of the course, or by popular accounts of how hard the course is, or by entry requirements.

5, Complexity, in both senses described here – the number of elements and the number of relations among them – is captured by Biggs in his SOLO taxonomy.

6, Scale is also captured by SOLO, and may be made manifest by the length of the reading list or the size of the textbook.

These close links between difficulties in abilities and difficulties in subjects, topics or ideas should not be a surprise. The many and complex relationships between abilities and knowledge are topics for many other blog posts. For now, I’ll simply note that, in education, knowledge usually embraces ability, which may be the ability to construct, critique, use and do many other things to and with the knowledge. Learning outcomes, on a good day, provide a bridge between knowledge and ability.

Beyond the six suggested above, there are further possible elements of difficulty in academic subjects, topics or ideas, if we work with learning outcomes rather than with knowledge as slabs of content.

7 Bloom offers us one way to conceptualise the difficulty of tasks. He considers a range of things we can do with and to knowledge, from recalling it, making sense of it and applying it to analysing it, evaluating it, and synthesising new knowledge. The higher levels of Bloom’s taxonomy are widely treated as corresponding to higher levels of difficulty, with for example lower level courses being confined to the lower levels of Bloom. I have serious difficulties with this, as i explore in other posts. But nonetheless the relating of Bloom level to difficulty is widely practised.

8 If we put Bloom and SOLO together, or rather set them orthogonal to each other, we can develop a useful account of difficulty as a composite function of level and complexity.

9 Wickedness is another dimension of difficulty, in the widely used sense of wicked problems. Wicked problems are not so much difficult to solve as insoluble, certainly incapable of an optimum solution. We may take solvability as another dimension of difficulty, applied of course to problems rather than just to content. To what extent, how far, can an exact or at least a satisfactory solution be obtained? The less solvable, perhaps, the more difficult.

10 If I can do arithmetic on small numbers, I can probably, with some care and effort, become able to do the same arithmetic on larger numbers. I am simply extending my capabilities.
By contrast if I can confidently do arithmetic and I am suddenly faced with differential and integral calculus, it may take me a long time to become competent and confident with them. They may feel to be much more difficult than doing arithmetic with larger numbers.

Why the difference in difficulty?

Doing arithmetic with larger numbers is still doing arithmetic. It’s more difficult, but it’s the same kind of activity. Differential and integral calculus are, by contrast, something else entirely. We have to change our view of what constitutes mathematics, adding calculus to arithmetic, and thereby add new ways of thinking, make sense of and then use new notations and concepts.

In Piaget’s language, calculus requires us to accommodate, to change, our view of what comprises mathematics. By contrast we can simply assimilate arithmetic with larger numbers into what we already knew about arithmetic. Rogers tells us that we are much more likely to resist new ideas and concepts that threaten our sense of self, our current worldview. Changing and extending paradigms, worldviews, is likely to be much more difficult than working within and extending current paradigms. This gives a further plausible account of difficulty.

11 Land and Meyer’s accounts of threshold concepts – ideas which among other qualities require a (usually irreversible) change in world view, and are therefore sometimes troublesome, difficult – usefully add to the work of Piaget and Rogers.

“What do you mean by difficult?”, or “Why do you say this is difficult?”, may be useful questions to ask learners, including of course ourselves. As the technology becomes better and better able to enact simple abilities, ideas, concepts, subjects, questions and tasks, then we are going to be left with the more difficult ones. (To which I reply, yippee, bring ’em on!)

Analysing the nature and sources of difficulty, as I have started to do here, doesn’t magically enable us to deal with these more difficult tasks. But it will give us valuable clues, precursors for planning and acting.

Notes
1 I am very grateful to Gerard Long, Head of Accounting and Finance at Waterford Institute of Technology, for a conversation earlier today which prompted this post.
2 The rather narrow broadband here in Jos, Nigeria, and the relatively small screen of the iPhone, together deny me ready access to references and sources. I shall insert these when back home, but not immediately,

As always – tell me what you think.