Do you understand it or just know it?

In English there is no word for ceasing to understand something (“ununderstanding”) but there is of course a word for ceasing to know something i.e. ‘forgetting’.  This points to something vital about the process of learning that is very often lost by both students and teachers – that the foundation of deep learning is understanding, not knowledge.  This is because understanding is a usually permanent gain in insight, whereas knowledge is often temporary and fragmentary.

But what’s the difference?  And why does it matter?   Here’s an example that may help to illustrate the point.  In A level maths you may be taught that there is something called the discriminant (b² – 4ac) which can be used to determine the number of roots of a quadratic equation.  (If the discriminant is positive it is has 2, if equal to 0 it is 1, and if negative it has no real roots).  I once had some students that knew this but did not know why.  I explained to them that the formula for solving a quadratic contains the discriminant (highlighted in red):

x = [-b ±√(b² – 4 ac) ] ÷ 2a

And then I asked them:  Can you square root a negative number?  To which the answer was no.  So then I asked them:  Well if the discriminant was negative, would that formula give you any answers?  And then I  saw the PENNY DROP moment!  And then I said:  And what if the discriminant was equal to 0, what would happen then?  And one answered:  “Well then it would just be √0 so x would just equal -b ÷ 2a!  So just one answer.”  And then of course they could answer for themselves what happens when the discriminant is positive.

This illustrates what I believe is central to effective learning:  that students not only knowing what is true, but vitally, WHY it is true.  It’s clear to me that those students will now never forget the meaning of the discriminant, because they can’t ‘ununderstand it’.  What’s more the understanding they’ve gained would lend itself to being able to handle other more complex situations.  For instance they would be able to use this insight to answer questions of the form:  “Prove that quadratics of the form kx² – 2kx + k  always have exactly one root.” Whereas students who only ‘know’ the discriminant might well struggle.

So my job as a tutor is always to inculcate understanding in my students, not simply knowledge.  And for students, your aim should always to be to understand what you are being taught, not simply to remember it.

What’s so good about tutoring?

Jeremy Wilburn / Foter / Creative Commons Attribution-NonCommercial-NoDerivs 2.0 Generic (CC BY-NC-ND 2.0)

Do you really need a tutor?   Maybe your teacher at school is very good and you are very bright, so perhaps you can get the grades you need without the help of a tutor?  Certainly some students go through their school years without any personal tuition and still get very high grades.  However in my view, in the UK, this is now a small minority.  But why is this?  What’s so good about tuition?  And why is it worth the trouble and expense?

In my view there are some aspects of tutoring that can make it invaluable in helping students achieve their potential.  In essence, personal tuition is the foundation for deep understanding, and true mastery of a subject.  As such it is no surprise that it forms the basis for teaching at places like Oxford and Cambridge!  Personal tuition has some huge benefits when compared with other methods of learning.  These are the main ones:

  • The tutor is able to meet the exact needs of the student, and pitch the questions and tasks at exactly the right level.  So, unlike at school, the student need never feel ‘left behind’, or alternatively bored and unstimulated.
  • The tutor can focus on those areas where the student needs most help.  A good tutor will be able to identify and concentrate on those areas causing the most trouble, so maximum progress is made in a minimum of time.  In school students often find that too much time is spent on some topics and not enough on others.
  • Tuition enables the tutor and student to form a close relationship based on mutual trust and respect.  This trusting relationship then allows the student’s barriers to learn (distrust, anger towards authority, fear of failure) to be recognized and removed.  In school these barriers are usually ignored and teachers are unable to form such relationships with their students.
  • Tuition ensures that students show they understand. If tuition is effective the student will be constantly thinking, answering questions and solving problems (the foundation for understanding and long-term memory) under the close watch of the tutor, rather than simply ‘hearing the teacher speak’, which too often occurs at school.  This means that while at school students often think they understand what has been taught, in tuition they will know they understand.
  • Students always have the opportunity to ask tutors the questions that are bothering them.  In school environments this is often not possible, due to the number of students and constant need to get through the syllabus.
  • Tuition encourages students to take responsibility for their learning.  Effective tuition encourages the student to consider what they want to learn from the tutor and why they are facing problems in a subject.  But at school students are encouraged to see learning as the ‘teacher’s job’, which undermines this sense of responsibility.

Don’t get me wrong, schools are not all bad, and of course not all tutors are good!  But, done right personal tuition can be a truly transformational experience for students.

Recommended reading for Economics students

♔ Georgie R / Foter / Creative Commons Attribution-NoDerivs 2.0 Generic (CC BY-ND 2.0)

Economics AS students have now finished their exams and the pro-active among them will be starting to look ahead to next year and their applications to university.  At this time of the year I am often asked for recommendations for wider reading.

Whenever I am asked this I think there are two main aspects of Economics that make it a fascinating subject to teach but also which students should be aware of when considering it as a choice of degree subject.  The first is that  it is a varied and controversial subject with a number of continuing disagreements (for instance over the nature of capitalism).   The second is that the subject overlaps with a number of other subjects including Politics, Philosophy, History, Psychology, Finance, Mathematics, and Statistics, to name just a few.

So here are 5 choices for the ambitious student which attempt to reflect these ideas.

The Communist Manifesto, Karl Marx.  Seemingly obscure, or out-of-date this is still one of the best and most insightful critiques of capitalism ever written.  The language is occasionally off-putting to students but written with a rare combination of passion and clarity.

The Affluent Society, JK Galbraith.  For most of human history the problem confronting human beings has been one of scarcity.  But for some countries, the new challenges are the problems associated with affluence and abundance.  Galbraith – always one of the best economic writers – highlights the problems of waste and misallocation of resources present in our own societies.

Capitalism and Freedom, Milton Friedman.  The Anti-Marx!  Friedman defends capitalism against its attackers and argues that it’s the basis for all important freedoms.  Not only that but it should be extended while government involvement should be reduced.

The Wealth and Poverty of Nations, David Landes.  A historical overview of the success of capitalism in generating wealth and development.  Fascinating historical insights informing a pro-market outlook.

Mathematics for Economics and Finance, M. Anthony and N. Biggs.  The use of mathematics in economics is frequently unnecessary and off-putting but it’s an unavoidable part of the subject.  Anthony and Biggs make it very accessible, while always explaining the purpose of the models used (as well as their limitations).

The dangers of studying too hard

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The title of this post might seem a bit odd, after all surely most students are not working hard enough!  Well, not usually!  While there are many students who would benefit from working more, in my experience a large proportion are actually working too hard and this is one of the reasons why they are doing less well than they could.  In fact I would go as far as to say that one of the main reasons students under-perform in their exams is that they have worked too hard.

Why is working too hard bad for you?  Primarily because it makes you tired.  As you get more tired your ability to learn falls significantly.  A common pattern is that overworking leads students to struggle to concentrate in lessons and when doing their homework.  As their performance drops they then (wrongly) conclude that they need to work more and so the cycle continues.  Usually this only ends when the student falls ill and has to take time off school.  That time-off then leads the student to fall further behind, causing even more difficulties.

Another reason working too much can be a problem is that when you work too hard this is usually at the expense of other important activities like spending time with your family and friends, relaxing, doing sport or other hobbies.  While they can sometimes seem like an optional luxury, in fact they are crucial  to your health and happiness and should not be neglected.  And while doing those vital things your mind assimilates the information you have learned and your body recharges its energy stores.  When those activities get dropped the tendency is to feel more tired, bored, and irritable which is not conducive to working productively.  Also the more you work, the more your life is centred round work which can make it seem all important and very stressful.

So how do you know if you’re working too hard?  These are some important signs to look out for:

– You struggle to concentrate during lessons or homework

– You sleep poorly and wake-up feeling tired and sluggish rather than refreshed

– You find yourself feeling unmotivated or lacking interest in your work

– You keep making the same mistakes in your homework or exam practice – your grades are not improving

– Homework seems to take you several attempts and much longer than it takes others

– On the weekend all you want to do is sleep

– You frequently get colds and flu

If you are experiencing several of these then WATCH OUT you could be approaching exhaustion or burnout, meaning you will need to take time off school / college to recover.

So what do you do about this?  How much should you work?

No one can say exactly how much or when you should work, as it varies enormously from person to person.  However there are a few principles you can follow which should help prevent this problem:

Don’t work for longer than 1 hour at a stretch and even better, 40 minutes.  Then have a break where you do something that doesn’t tax you like go for a walk or have a snack before re-starting.

Few people can cope with studying for more than 6 hours in total in one day.  If you have also been at school then it is probably 2 hours in one evening.

As you approach exams you should be steadily reducing the number of hours you work as you approach the exam, not increasing it!  This will ensure that you are fresh for your exam.

If you have a revision timetable you should also block in time for fun, exercise,  relaxation and treats which you look forward to.  Don’t cut these out!

– If possible, don’t study in your bedroom as this can make the room associated with work and stress, making it harder to sleep well at night.  (If you have to work there then make sure you pack away books and notes at the end of the day so that it feels finished before trying to sleep).

If you commit to these things what you will find is that your work is no longer the chore it once was, that you come to it refreshed rather than tired and that you see that work is part of life, not all of it!  You will work much more productively and achieve better results.   Sometimes less really is more!

How to be a better student

JuditK / Foter / CC BY-ND

At this point in the year students seem to put their heads down to focus on understanding all the difficult new topics they need to learn.  Many have exams in January which they are anxious to do well in.  But more often than not the barriers facing students are not within the subject they are learning but within themselves!  Focusing on addressing those can yield benefits across all  subjects and significantly improve grades.  The following are the habits I have observed in my most successful students, regardless of their subjects.  How many are yours?!

Taking responsibility for both their achievements, and failures

This is the single most important habit, but the one which can be most elusive.  When things go well most of us are happy to take the credit and bask in the approval of others.  But when they don’t go so well we generally seek an excuse:  “I wasn’t feeling well”, “The teacher is rubbish”, “It was a tough exam”.  But the danger of this is that we miss a major opportunity to learn and adapt our approach in order to improve in the future.  If you find yourself reading this and thinking “Nah, that’s not me” then maybe just pause for a moment!

Highly effective students always know that they are responsible for their own learning, successful or not.  They don’t look for excuses but instead seek ways of changing their approach and understanding things better.  I once had a student who repeatedly made small errors in algebra which would lead to her dropping a lot of marks in maths exams.   For a long time she ignored the problem and pretended it was because her teacher was inadequate. When she eventually accepted that this was her responsibility it quickly stopped and her marks improved hugely.

So use failures and disappointments as a launchpad for future success!

Always wanting to understand, not merely remember or reproduce 

The problem with lots of teaching and learning is that the teacher and the student remain content with the ability to reproduce the material covered in class, rather than for the student to truly understand why it is true.  But it is only understanding that will enable you to answer the unpredictable questions you will face in the exams.  There is no word for “ununderstanding”  because it doesn’t exist!

So when in class or reading a textbook always ask: “What does this mean?”, “Why is it true?”, “How does it connect to other things I know?”  Challenge your teacher or tutor to answer these questions!  Challenge yourself!  Use all the resources available to you to get you to a state of deep understanding.  Then you will be able to tackle anything thrown at you and you won’t have to memorise anything (well almost)!

Always reflecting on and learning from experience

We all make mistakes.  They are a fact of life.  And in fact they are a crucial means by which we learn.  But if they are ignored or their lessons are disregarded then we risk repeating the same mistakes over and over again.  I once had a student who kept sitting exams and running out of time, so his marks were never as he should have achieved.  We discussed techniques he could use to improve this but then they would just be forgotten about or dropped and he would revert to his old ways.  So he never improved.

If an exam or test doesn’t go as well as expected, the successful student will always ask themselves why, and seek a clear answer to that from their teacher or tutor if necessary.  They will then act on the answer.  So if the answer is “I don’t understand conditional probability” they will then go away and make sure they work on it until they do understand.

Being organised

You can be highly intelligent, motivated and hard-working, but if your notes are not filed properly, you are always late for your classes and you don’t know what’s in the syllabus then you are not going to achieve much.

Getting organised is a matter of being in control of your studies rather than letting it control you.  It means:  always being on time for lessons and never missing deadlines, keeping your notes organised and accessible, always knowing what you need to know and by when.  It means doing this not just periodically when it has all got out of control but always, as a matter of habit.

Knowing how to rest, relax and enjoy themselves

If this is all sounding like hard work then it’s important to emphasise that the most successful students are not necessarily those that spend the most hours studying.  The ones that spend the most time are usually wasting their time because they are disorganised, bored or tired.

In fact successful students  know and appreciate the value of time spent playing sport, resting, or just having fun.  They know that this time is not only valuable in its own right but also provides time-off for the learning mind to recuperate and assimilate new information.  People forget that when you are not thinking your unconscious mind is still learning!  Time spent on hobbies and pastimes also helps put studies into perspective and avoid the stress and anxiety associated with over-work.    So it’s vital to build in regular times for rest, sports, social activities and fun in and around your studies.

Time spent developing these habits might just be the best thing you ever do!