Since writing part 1 of this series, I have been thinking about what I wrote, and discussing it with another teacher, and there are some aspects of what I said that I would like to expand on, clarify, or change.

Practice is an important part of learning

In my previous post I said that learning something involves both understanding and memorisation. For example, in order to learn the definition of an ecosystem you have to memorise the wording of the definition and understand what it means.

However, I now realise that by focusing too much on certain types of learning tasks, I was missing another important part of many learning processes: practice.

For example, let’s say that someone is learning how to throw a basketball into a basketball hoop from the free throw line. This learning process is likely to involve some understanding and memorisation, because the person has to understand what the correct technique is and memorise the movements that they need to do with their body. However, for most people the bulk of the learning process is not going to consist of memorisation and understanding, but of practice. They will need to stand on the free throw line and repeatedly try to throw the ball into the hoop until they get good at it.

It is not just physical tasks that involve practice. For example, many learning tasks in mathematics require lots of practice before they are mastered. In my experience, maths students often get to the point where they understand how to do a certain type of question, and have memorised the steps, but because they have not done enough practice they make lots of little mistakes and end up getting the answer wrong. By practising lots of the same type of question, they can iron out these mistakes and become good at that type of question.

Also, there are many academic tasks (especially outside of maths and science) which are not so much about getting a correct answer as they are about learning a technique. Essay writing, analysing texts, and producing artwork are three examples where there is no correct answer to be learned, but where a person competent in that technique can fairly easily judge a good example of it from a bad one.

Different learning tasks vary in the amount of memorisation, understanding and practice involved

Although memorisation, understanding and practice are three important components of learning, not all learning tasks require a significant amount of all three.

I have already given the example above of learning to throw a basketball into a hoop – a task which mainly involves practice. In general, learning tasks that consist of gaining a new skill (as opposed to gaining new knowledge) are likely to be predominantly practice based.

There are some learning tasks that mainly involve memorisation, with little or no understanding or practice required. For example, let’s say that a student is tasked with learning these Spanish words:

• Cebolla – Onion
• Espinaca – Spinach
• Frijoles – Beans

Learning vocabulary like this is essentially just a memorisation task. No understanding or practice is required.

Of course, it could be argued that in order to memorise the words you have to repeatedly test yourself on them, which is in fact practice. So in a sense, practice is involved in the memorisation process. However, this is a bit different to other tasks (like the basketball example) where practice is required not simply as a method of memorisation, but as a separate part of the learning process. (Also, technically, there are ways of memorising things which don’t involve practice – specially mnemonic methods, which can be used to memorise things without going through a process of repeated testing).

As you can see, the question of exactly how much memorisation, understanding and practice are involved in a particular learning tasks can end up coming down to definitions, and can also depend on the way in which the learning task is completed. However, I think the important point is that, regardless of their relative importance in particular learning tasks, these three processes are the main processes involved in learning.

Prerequisites are not always all-or nothing

In the previous post, I defined a prerequisite as something that must already have been learnt before something else can be learnt. In other words, if learning task A is a prerequisite for learning task B, then learning task A must be completed before learning task B can be successfully completed.

However, there are many situations where completing learning task A isn’t strictly required in order to complete learning task B, but it is more difficult (sometimes much more difficult) to complete learning task B if learning task A hasn’t already been completed. Or, to put it the other way round, completing learning task A makes the completion of learning task B much easier.

Let’s look at some examples:

1. Times Tables

In primary school, students are supposed to learn how to multiply two fractions. This is done by multiplying the tops of the fractions to get the top of the new fraction, and multiplying the bottoms to get the new bottom. For example:

Multiplying 6 by 3 gives us 18, and multiplying 7 by 8 gives us 56. (I have left out the simplification step which would usually be done next).

If I was going to teach a student to multiply fractions, I would first make sure that they have learnt their times tables (up to at least 10 x 10). Knowing your times tables means that you have memorised the answers to all of the multiplications between two one-digit numbers. Clearly this is very important prior learning for someone who is learning to multiply fractions. In the example above, a student who knows their times tables will know that 6 x 3 = 18 and 7 x 8 = 56, meaning that all they have to learn is the technique for multiplying two fractions (multiply the tops, then multiply the bottoms).

However, it is not actually necessary to know your times tables in order to learn to multiply fractions. As long as you have previously learnt some method of multiplying two numbers, you will be able to do it. For example, if you want to multiply 7 by 8, one way to do it is to add eight 7s together (7 + 7 + 7 + 7 + 7 + 7 + 7 + 7). However, clearly this is going to make the process of multiplying the two fractions much slower and more liable to errors than simply having the answer to 7 x 8 memorised.

(As a completely off topic side note, I frequently teach students GCSE and A-level maths students – including ones who are getting high grades – who don’t know their times tables or how to multiply fractions).

So we can see that, according to the definition I gave in the previous post, learning your times tables is not a prerequisite for learning to multiply fractions. However, for many practical purposes (e.g. designing syllabuses, planning schemes of teaching) we may want to treat it as one. Perhaps we could describe learning the times tables as a kind of ‘semi-prerequisite’ for learning to multiply fractions.

2. Reading Comprehension

In his 1976 book Human Characteristics and School Learning, educational researcher Benjamin Bloom argues that there might be certain learning tasks which are prerequisites for almost all of the learning tasks that students are expected to complete in school. Possible examples that he suggests include:

• Reading comprehension
• Basic arithmetic skills
• Logical reasoning processes
• Language development
• Writing competence
• Attention skills
• Study skills
• Use of a library
• Management of time
• Use of learning resources outside of the classroom

The one which he focuses most on is reading comprehension, since, as he points out, most courses in school tend to emphasise learning by reading textbooks. In Bloom’s own words:

“[T]he emphasis on textbooks in schools throughout the world has in all too many instances reduced each subject to the reading about the subject. Thus, the introductory psychology course is reduced to the reading of a standard textbook in the field.”

Benjamin Bloom, Human Characteristics and School Learning(page 68)

Of course, a good level of reading comprehension is not strictly necessary in order to complete most school learning tasks, since there are non-written sources of information available. For example, teachers can verbally explain things to students. And one advantage that today’s students have over those in 1976 is that they have access to a vast amount of online videos taking them through the content that they need to learn.

Nonetheless, it is still true that a large amount of the information available to students (especially when they are at home in the holidays revising for exams) is in written form, and therefore it is clearly true that students with good reading comprehension skills are going to have a much easier time learning the things they need to learn than students with poor reading comprehension skills.

Therefore, we could argue that learning reading comprehension is a kind of ‘semi-prerequisite’ for a wide range of other school learning tasks.

It is worth noting that this is not an inevitable result of the nature of the school learning tasks, but is more a consequence of the predominant ways that students are taught in schools. Indeed, Bloom goes on to say:

“It is possible for the introduction to a subject like psychology to be vastly different from a reading course in the field. Science courses may emphasise reading a textbook about science or they may emphasise observation, experimentation, and first-hand experience with scientific phenomena. The humanities need not always be the reading about art, literature, drama, dance, or music. There are probably more useful and more direct ways of dealing with these subjects than reading about them. Whether these other ways of dealing with these subjects are more valuable depends on what are believed to be the desirable content and objectives of these subjects or fields.”

Benjamin Bloom, Human Characteristics and School Learning (page 68)

Of course, the emphasis on textbook reading in all subjects almost certainly helps those students with poor reading comprehension skills to improve in this important ability. However, this is traded off with limiting their learning of the subjects themselves.

3. Related knowledge can aid memorisation

There are lots of cases where knowing certain things can make it easier to memorise certain other things. This is often because the prior knowledge, while not required, helps to give context to the new information, making it more meaningful and therefore easier to remember.

In one of the examples above, a student was required to learn that espinaca is the Spanish word for spinach.

In Spanish, words do not usually begin with ‘sp’. For many English words that begin with ‘sp’, the Spanish equivalent word begins with ‘esp’. Knowing this is clearly not necessary in order to learn that espinacha is the Spanish word for spinach, but for some students it may help. Understanding common ways in which words change when they are translated between your native language and the one you are trying to learn can make learning a language a lot easier.

When I am teaching my science and maths students, I often find myself going beyond what they need to know, in order to give some context to the things they need to know. Although it is a difficult balance, I often find that my students like this as sometimes learning slightly more can actually be easier than learning less, if the extra content helps to tie everything together into a more meaningful and easier to remember set of concepts.

So this kind of helpful, but not necessary, prior learning could probably also be described as a ‘semi-prerequisite’, but not as strong of one as the previous two examples.

Is There a Prerequisite-ness Spectrum?

Perhaps it could be useful to think of prerequisite-ness not as a binary thing but as a spectrum that looks something like this:

I think this model of prerequisiteness probably is more realistic than the binary one that I presented in the previous post.

However, for the rest of this series of posts, I think I will probably continue to talk about prerequisiteness as though it were binary.

The reason for this is that I don’t think it really makes a difference to the conclusions that I am building towards and I hope that anyone reading along will agree that if I had talked about it in a spectrum way the message would have ended up being the same, but the posts would probably be a lot longer and more complicated. So, for now at least, I am happy with the simplification of saying that A either is a prerequisite for B or it isn’t.

This is the first in a series of posts in which I’m going to set out how I think learning works. Nothing that I’m going to say is original, but hopefully I can bring together lots of things I’ve read and thought about and present them in a clear, simple way.

I’m not going to give any evidence for these ideas at the moment. Not because I don’t think there is any (actually I think there is a lot of evidence), but because I want to use this series of posts just to lay out a description of learning, and then in future posts I will discuss the evidence for that description. In other words, this is a description of how I think learning works, and in future I will explain why I think this.

Learning is gaining new knowledge or skills

All humans know things (for example, I know what the capital city of France is) and can perform certain tasks (e.g. I can add two digit numbers in my head). In other words, we have knowledge and skills.

We also have the ability to gain new knowledge and skills. This is called learning. For example, when I was born I didn’t know what the capital of France was and I couldn’t add two digit numbers – I have learnt these things. Go me.

Okay, so far it’s all pretty obvious. But I’m a teacher and I like to start with the basics and build up, so here we go…

Most learning requires you to have already learnt other things

If you have ever tried to explain something to someone, you probably know that in order to do so you have to assume that they already know and understand certain things. In other words, their ability to understand the explanation relies on them having previously learnt some other things.

If they haven’t already learnt those previous things, then any explanation that assumes they have is probably not going to work. You will be better off going back and explaining the previous things first. Of course, your explanation of the previous things may rely on them understanding some other things, so you may find that you have to go back several steps and then work forwards until you finally get to the thing that you were originally trying to explain.

This is what I do when I’m tutoring a student. I start from what the student already understands and work forwards. It’s also what I try to do when I’m teaching a class. However, with a class it can be much more difficult as the students in the class usually vary greatly in terms of which things they have already learnt. This is part of the reason why being at the front of the class explaining things is something that classroom teachers are moving away from.

If a certain learning task needs to be completed before another one can be, we can say that the first learning task is a prerequisite to the second. ‘Pre’ means ‘before’ and ‘requisite’ means ‘needed’, so a prerequisite is something that you need to do before you can do something else.

A set of learning tasks can be sequential or non-sequential

To say that a set of learning tasks are sequential means that they need to be completed in a particular order. In other words, the first learning task is a prerequisite to the second, which is a prerequisite to the third and so on.

For example, let’s say that a secondary school biology student needs to learn the following definitions:

1. A producer is an organism that makes its own food.
2. A primary consumer is an organism that eats producers.
3. A secondary consumer is an organism that eats primary consumers.

Learning definition 1 is a prerequisite to learning definition 2. This is because definition 2 contains the word producers. If the student hasn’t yet learnt definition 1 then they don’t know what a producer is, so they can’t understand definition 2.

Similarly, learning definition 2 is a prerequisite to learning definition 3, since definition 3 contains the phrase primary consumers and the meaning of this phrase is explained in definition 2.

Therefore we have a chain of sequential learning tasks that looks like this (an arrow means that one learning task is a prerequisite for another):

To say that a set of learning tasks are non-sequential means that they can be learnt in any order. An example would be learning the following biology definitions:

1. A habitat is the place where an organism lives.
2. Predation is when one animal eats another animal.
3. Herbivory is when an animal eats plants.

These definitions can be learnt in any order and therefore the learning tasks are non-sequential. We can represent them like this (without arrows between them):

Learning involves both memorisation and understanding

Let’s have another look at the first set of learning tasks discussed above:

1. A producer is an organism that makes its own food.
2. A primary consumer is an organism that eats producers.
3. A secondary consumer is an organism that eats primary consumers.

It could be argued that these tasks are not sequential at all, because in order for the student to pass their upcoming biology test all they have to do is memorise the words of each definition and that can in fact be done in any order. For example, if you take the phrase ‘learn definition 3’ to mean ‘memorise definition 3’ then you don’t actually need to know what ‘primary consumer’ means in order to learn definition 3. In fact, you don’t need to know what any of the words in definition 3 mean in order to learn it.

This touches upon the important distinction between memorisation and understanding, both of which are important parts of learning. Many sets of learning tasks that are sequential from an understanding point of view are in fact non-sequential if all that is required is memorisation.

In fact, UK school students regularly memorise things which they don’t yet have the prerequisite knowledge to understand.

For example, I have taught many GCSE and even A-level maths students who have managed to successfully memorise the steps to factorise a quadratic expression despite not knowing what a quadratic expression is, or having a grasp of the basic concepts of algebra, or in some cases even basic arithmetic.

Since these students don’t yet have the prerequisite knowledge and skills to be able to understand how to factorise a quadratic expression, before I can teach them to understand it I need to go back to what they do understand and work forwards gradually from there. When I do this, it is always met with an exasperated response from the student who will inevitably say something like, “why do we need to do this, when I already know how to do it?”. From their point of view they already get the right answers to the questions, so no more learning is needed.

Understanding and memorisation are both needed

In a way, I can understand where my students are coming from when they say that they have finished learning something once they have memorised it. It is possible to memorise a lot of things without actually having the prerequisite knowledge to be able to understand them, and since a lot of exam questions can’t distinguish between memorisation and understanding, you can actually get quite far just by doing this.

However, there are two problems with this type of learning.

Firstly, if you continue studying the subject to a higher level, you will eventually get stuck. At some point you will need to apply your understanding of the things you learnt previously and you won’t be able to do that because you don’t have any understanding to apply.

The second, and arguably bigger, problem is that memorising a bunch of stuff you don’t understand is pretty pointless. I mean, it can help you do well in exams and therefore get good qualifications and therefore get a well-paid job etc. So from that point of view it is useful.

But as a society, do we want to spend lots of time, money and effort teaching young people to memorise loads of things that they don’t understand and will never use? Or do we want to teach them to gain a deeper understanding of the world around them? I also know children don’t actually want to learn things that are pointless. After all, every teacher is all too familiar with the question “what’s the point of learning this?”.

I should be clear that I am not against memorisation. In fact memorisation is a very important part of learning and I spend a lot of my teaching time helping my students to memorise things. But you need both. You won’t be a very good ecologist if you can’t give a definition of a simple term like habitat, but you also won’t be a very good ecologist if you don’t understand what that definition means.

Therefore, from here on I will use the word ‘learn’ to mean ‘understand and memorise’. I defined a prerequisite as something that must be learnt before something else can be learnt, so in these terms a prerequisite means something that must be understood and memorised before something else can be understood and memorised.

Learning usually involves a mixture of sequential and non-sequential learning tasks

So far we’ve looked at examples of just a few learning tasks where they were either all sequential or all non-sequential. Of course, the real world is rarely that simple and we usually find a mixture of the two within an set of tasks.

For example, let’s say our biology student has been given the following definitions to learn:

1. Biotic factors are ways in which an organism is affected by living things (e.g. predators, prey, disease).
2. Abiotic factors are ways in which an organism is affected by non-living things (e.g. temperature, humidity, rainfall).
3. A population is all of the individuals of one species living in a particular place.
4. A community is all of the populations of all the different species living in a particular place.
5. An ecosystem consists of a place, the community of organisms living in that place, and all of the biotic and abiotic factors that affect them.

Let’s think about which of these are prerequisites for which others:

Learning definition 3 is a prerequisite for learning definition 4. This is because definition 4 refers to populations and the meaning of this term is explained in definition 3.

Learning definitions 1, 2 and 4 are prerequisites for learning definition 5. This is because definition 5 refers to biotic factors, abiotic factors and a community – terms whose meanings are explained in definitions 1, 2 and 4.

There are no other prerequisites within this set of learning tasks (of course some of them have prerequisites which are outside of this set of tasks – e.g. some use the words ‘organism’ and ‘species’ which need to be understood before these definitions can be).

We can represent the relationships between these learning tasks as follows:

We can apply the concepts of sequential and non-sequential at different scales

So far, we’ve looked at examples of small learning tasks – learning short definitions. We can also group lots of small learning tasks together into bigger learning tasks.

For example, courses are usually divided into a number of parts, usually called ‘modules’, ‘units’, ‘chapters’ or ‘sections’. Each of these parts usually contains several things which need to be learnt. So we could think of ‘learn module 1’ as a learning task which is made up of lots of smaller learning tasks.

Just like the smaller ones they are made up of, these bigger learning tasks can be sequential or non-sequential. For example, a GCSE chemistry course might contain the following modules:

1. The structure of atoms
2. Bonding between atoms
3. The reactions of metals
4. The reactions of oxygen

Before you can understand bonding between atoms, you need to understand the structure of the atom, so learning module 1 is a prerequisite to learning module 2.

An understanding of bonding is needed to be able to understand any kind of chemical reaction, so learning module 2 is a prerequisite to learning modules 3 and 4.

However, it does not make a difference whether you learn about the reactions of metals first or the reactions of oxygen first, so learning module 3 and learning module 4 are non-sequential learning tasks with respect to each other.

In other words, the relationship between the learning tasks looks like this:

Summary

Learning is the process of gaining new knowledge and skills.

Most learning tasks have prerequisites – other things that must be learnt (i.e. memorised and understood) before the current task can be learnt.

A set of learning tasks where each one is a prerequisite for the next is described as sequential. A set of learning tasks without prerequisites is described as non-sequential. Usually, a set of learning tasks is a mixture of sequential and non-sequential. The concepts of sequential and non-sequential can be applied at many different scales.