We all want our students to *understand* what they learn, right? I can’t think of any teacher that does not. However, the idea of teaching for understanding can be problematic and lead us into some serious, and highly popular, misconceptions.

In this post, I will outline what understanding actually *is* before explaining the best way that we can promote it.

**Rote learning**

Many teachers are keen to avoid ‘rote’ learning. This is when students memorise facts and procedures without really understanding why they are doing it. It is possible to do this, especially if you practice something a lot like a mathematical procedure. However, memorising facts that have no meaning to you is extraordinarily difficult. This is why memory champions who can memorise the order of a deck of cards, for instance, tend to impose a spurious meaning on it. They may imagine walking through a house and seeing a different card in each room.

We also do this in education. At school, I learnt the order of the colours of the rainbow by memorising, “Richard Of York Gave Battle In Vain.” Unfortunately, I wasn’t taught history properly and so the resonances that the writer of this mnemonic intended were rather lost on me.

So, by repetition, you *can* learn the ‘how to’ of something like a mathematical procedure without learning the ‘why’. And you can also learn key facts this way such as times-tables. However, anyone genuinely attempting to teach an entire curriculum in a rote fashion would find the task impossible. Those who maintain that standardised tests are largely tests of rote learning are likely wrong.

**The nature of understanding**

I think that one problem arises when we start to see understanding as something qualitatively different to knowledge. Don’t mistake me here, understanding is a useful concept and I am not trying to replace it but we need to recognise that it is the degree of a thing rather than a thing in itself. In their influential book “Understanding by Design,” Grant Wiggins and Jay McTighe suggest that pieces of knowledge are like tiles and understanding is like the pattern that the tiles make. This is useful as far as it goes and brings to mind the psychological concept of a ‘schema’ where related ideas are organised in the mind and therefore understanding represents this organisation. However, I suggest that a fractal is a more apt analogy; it’s tiles all the way down.

I have written before of two types of understanding; objective and subjective. Objective understanding is where we perceive that someone else understands a concept. In his book, “Intuition Pumps And Other Tools For Thinking,” the philosopher Daniel Dennett contrasts a small child saying, ‘my daddy is a doctor,’ with a young adult saying, ‘my dad is a doctor’. He draws this comparison for different reasons to me but I would point out that we would perceive the small child to understand this statement less than the young adult. What accounts for the difference? Well the young adult will have a lot more relevant world knowledge; she will know that doctors work in hospitals or clinics, that there are different kinds of doctors, that doctors are relatively well paid and so on. So her deeper understanding actually consists of a larger amount of relevant knowledge.

Subjective understanding is the feeling that you understand something. This can be quite deceptive. However, assuming that the feeling is accurate, what accounts for it? Well, let’s look at the opposite; a feeling of confusion. This can be easily induced by presenting a student with a complex problem with many elements to attend to. However, if the student recognises many of the elements because she already has ideas about them available in long term memory then the confusion will be diminished. Sufficient relevant knowledge will mean that the student feels that she understands the problem. Again, understanding is effectively an accumulation of relevant knowledge.

Further, this can be facilitated by some procedural knowledge. If you can automatically perform the ‘how to’ part of a mathematical problem or immediately recall the answer to “6 times 8” then you don’t have to devote attention to these parts of the problem and will have a better chance at apprehending the ‘why’. As Alfred North Whitehead suggests:

*“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”*

**Dangerous Knowledge**

Unfortunately, as Whitehead suggests, many people draw precisely the opposite conclusion about understanding. In their famous study, Kamii and Dominick conclude that directly teaching mathematical procedures actually *gets in the way *of students understanding the maths. These studies are not perfectly controlled experiments and an attempted replication (of sorts) by Stephen Norton in Queensland found the opposite result.

This is also conventional wisdom in science education. Teaching ‘facts’ is harmful to understanding and should be replaced by students engaged in activity and finding things out for themselves.

The logic seems to be:

1. Traditional approaches to teaching leave students with lots of misconceptions

2. ‘Constructivist’ teaching approaches are an alternative to traditional approaches

3. Therefore, we must use constructivist teaching approaches

This is a classic non sequitur. How do we know that the constructivist approaches will work any better? The evidence in favour of them is extremely thin. It follows the logic of the politician’s syllogism:

1. We must do something

2. This is something

3. Therefore, we must do this

**Misdirection**

Unfortunately, this sort of thing sends us off on a wild-goose chase. Instead, we should be looking at how we can improve conventional teaching approaches so that we can reduce misconceptions and maximise students’ levels of understanding. Interestingly, an important difference between experts and novices appears to be the ability to perceive the ‘deep structure’ of problems and situations. There are two key strategies to consider if we want to develop this appreciation of deep structure.

Woefully neglected in the research, teacher explanations are pretty critical to developing student understanding. I wrote about this for the TES. Unfortunately, it is not clear exactly how we can structure explanations in order to make them optimal. We can perhaps infer a few principles from the second strategy.

Building episodic knowledge seems to be effective – ‘episodic knowledge’ just means students’ memory of encountering similar problems or situations. This involves running students through many examples with the same deep structure. In physics, this might mean teaching a concept or principle, giving an example, asking students to complete similar examples before gradually and progressively diversifying into other problems with the same deep structure but different surface features. Once a number of different principles have been grasped then students can be presented with problems with different deep structures to see if they can identify what this is in each problem.

This suggests also that explanations should not just focus on the principles but also highlight the deep structure in different problems and situations, perhaps comparing these situations. In mathematics, you may wish to move through multiple representations e.g. from a rule to a graph and back again. The point is that the deep structure is retained between these representations whilst the surface features change, again allowing students to better grasp this structure.

We can’t bypass this stage. It appears that it is quite natural for new learning to be locked to surface features initially and it takes a lot of hard work to move past this.

**We don’t have to choose between knowledge and understanding**

It is both misconceived and potentially harmful to conceive of understanding as a spooky kind of a thing that is *qualitatively different *to knowledge. It leads us into thinking that gaining knowledge interferes with gaining understanding whereas the reverse is the case; the two are mutually supportive. And it leads us away from more effective approaches for developing understanding.