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.

Reblogged this on The Echo Chamber.

Great post. Some really important points.

A couple of comments; one about rote learning and the other about episodic memory.

Rote learning: In the heyday of ‘traditional education’ paper was expensive, so many new state schools couldn’t afford it. They couldn’t afford books either. Consequently, children wrote on slates and learned by rote entire lessons read out from a book by the teacher. Their ability to recite the lesson from memory was tested by monitors. High stakes assessment took the form of tests of the 3Rs and the results formed the basis for a teacher’s performance related pay.

Not surprisingly, the emphasis was on memorisation, so explanation often fell by the wayside. Many of the early objectors to rote learning weren’t objecting to learning things off by heart as such, but to the assumption that if a child could recite a lesson, they must be able to understand it. These objections to rote learning entire lessons were later extrapolated to all rote learning, and eventually (wrongly) to the concept of rote learning itself.

Episodic memory is a term usually used to refer to *particular* episodes. It’s powerful because the memory is accessible via several different associative routes; it includes a range of sensory and emotional links. The surface features of a problem will be more vivid to students than the deep structure because the surface features (e.g. the dimensions of a pool table vs the time taken to mow a football pitch) will tap into their previous (episodic) experience. That’s why students tend to get distracted by the surface features and often don’t spot deep structure similarities spontaneously.

Completely agree that deep structure needs to be taught explicitly. Drawing a diagram of the problem is a very powerful way of representing the deep structure and minimising the distraction of the surface features.

With regard to memory champions and decks of cards, there is a different way of doing this now. Each card is represented by a person, a place and a verb, so the King of Hearts might be Bill Clinton in the Whitehouse, smoking and the Queen of Diamonds might be the Queen at Buckingham palace knighting, and the Jack of clubs might be Cliff Richard at Royal Albert hall, strumming. So if those three cards in order you have Bill Clinton at Buckingham palace strumming. So you can do a deck in 18 images. Shows the power of knowledge!

I have always considered that deeper understanding is in some ways connected to a richer set of experiences, as much as greater knowledge. Is that worng?

I would suggest it is problematic if it leads to people downgrading the role of knowledge and not teaching enough of it.

… ‘wrong?’

Dunno. I suppose that my description of building episodic knowledge could be described as a ‘rich’ experience and so it depends what you mean. Certainly, if you are suggesting that understanding is a different kind of thing to knowledge then we differ.

I suspect that episodic knowledge and experience mean the same or very similar to me. I might be wrong but doesn’t Graham Nuttall’s work suggest that one of the key elements of learning is tackling knowledge in different contexts?

Whether those must be domain specific would be another question.

The primary step to depth of understanding is knowledge acquisition. Teachers have to explain knowledge about their subject but give opportunities for a student to broaden this through wider reading and wider discussions. However many tend to make experience and discussion the means to knowledge. The problem there is understanding remains superficial.

If your saying that that there is a link between ineffective teaching and a focus on planning activities over planning learning then I would agree.

Reblogged this on Dr Mike Beverley.

Reblogged this on teaching knowledge and creativity.

Conceptual understanding and “procedural” understanding (i.e., the mechanics of the operation) often work hand in hand, so that a student may first learn the procedure and then after practice come to understand the conceptual underpinning. A teacher may explain, for example, the meaning of place value and show how it works when subtracting with borrowing (now called “regrouping”). The student may have a vague understanding of what’s going on but can do the steps. With practice, and then another exposure to the explanation, the student may then grasp what is going on conceptually. Others may get it before mastery of the procedure. Still others may never really understand it but can function just fine with the procedure. Skills without understanding permit functioning and being able to solve problems. Understanding without skills, however, does not allow such problem solving ability.

Similarly, I never learned the reasoning behind invert and multiply for dividing fractions, except in the easier case of whole numbers divided by fractions. But a fraction divided by a fraction (e.g., 3/4 divided by 5/9) wasn’t explained. I learned it eventually when I knew enough algebra to be able to represent what was going on. But until then I knew other aspects of what was going on; i.e., my understanding included what is happening when one divides by a fraction, and when to use it to solve problems. (I.e., how many 3/8 inch intervals are there in 31/32 of an inch, means you divide 31/32 by 3/8). That I couldn’t explain why the invert and multiply rule worked was not an indicator of my understanding of what fractional division was.

Understanding is often given more emphasis than it needs at times. The mentaility of “students must understand or they will fail” seems to be the prevailing motto of math reformers. I believe Ashman’s thesis that seeing the “deep structure” in similar problems is a saner way to approach “understanding”

‘I believe Ashman’s thesis that seeing the “deep structure” in similar problems is a saner way to approach “understanding”’

These ideas are much researched, much written about and much discussed. Deep structure is not a new concept by any means.

I have not claimed that it is. I was first introduced to deep structure here:

http://www.aft.org/periodical/american-educator/winter-2002/ask-cognitive-scientist

An important psychological approach to the question of understanding in math education is:

Stellan Ohlsson & Ernest Rees (1988). The Function of Conceptual Understanding in the Learning of Arithmetic Procedures. Cognition and Instruction, 8, 103-179. http://www.tandfonline.com/doi/abs/10.1207/s1532690xci0802_1?journalCode=hcgi20

The article is based on this report, available online:

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