G'day.
I wish to express a huge 'thank you' to those of you subscribing to my channel.
I see this as something of a partnership and your subscribing to my channel encourages me
and I trust that the videos I continue to provide
are going to be of use to you.
This particular video is a response to a particular subscriber
who contacted me a week or two ago and asked,
"How do you memorise formulae in mathematics?"
Well, I realised, when I received the message, that I hadn't really explained that very well
in my 'How to Study' videos.
So, wikimaper, this video is for you ...
and for anyone else out there who wants to learn how to memorise formulae.
G'day.
Welcome to Crystal Clear Mathematics where it is easier than you think.
I'm your host, Graeme Henderson.
In this video we're going to learn some of the principles for memorising formulae.
Now, I don't pretend to be an expert.
I haven't made a life study of memory or psychology or neurophysiology.
I dabble a little in each of those areas but, primarily,
I'm a mathematician and a teacher.
So, I'm going to share my personal experience in learning when I was at school
and what I've discovered in sharing with students that I tutor.
Principle Number One is to learn with focus and intention.
That is, you CHOOSE to learn.
And what it means is that you set time aside
and you remove yourself from distractions, particularly electronic distractions ...
anything that will draw your attention, visually or audibly, away from what you're doing.
The benefit of doing that is you will learn faster and better and in shorter time!
What I propose to my students is that it's far better
spending five minutes' intensive time learning something well
than twenty minutes wasted time not learning it well at all.
I'd much rather spend five intensive minutes and then do fifteen minutes
of something I enjoy ... enjoy differently :-)
The Second Principle of memorizing is that it should be memorable.
It's interesting that we use that word.
It should be something that captivates our attention.
Now, it captivates our attention because we choose to focus on it ...
but people do this in a variety of ways.
There are some very, very good videos on YouTube (and some not-so-good ones)
where people show how to memorise, e.g., the quadratic formula in song.
Some people use a melody, that is, a very familiar tune
(or even a nursery rhyme tune) to recall the formula.
Others use rhythm ... for example, rap.
Some people use a story line and symbology.
And, if you watch a few videos on YouTube about how to develop your memory,
you'll find there are people who talk about the 'memory palace'
and things like that that can be extraordinarily useful as well ...
all different ways of using your visual imagination or your sense of rhythm or music
or some familiar tune or something familiar to you in order to establish this formula in your mind.
Another way of establishing it well is to link it to something familiar.
Academically, that means you can link it to something historical
or you can link it to mathematics that you already know
(which is my preferred way of learning) by learning the derivation of the formula.
That is, going from mathematics you understand and know
to the new mathematics and the new formula.
The Third Principle is that, in order to establish all this in your memory,
it has to be rehearsed intensively in the first week or so.
We call this 'massed practice.'
What I would normally do with my students, and what I did myself at school,
was I would derive a formula once or twice every afternoon for a week
or maybe two weeks, and memorise the formula very rapidly in that way.
And, the second thing I'd be doing, would be using it
and establishing its use in my memory.
The Fourth (and last) Principle is that, in order for that knowledge that you gain
in that intensive week or two to stay with you for months
and months and months until you sit for a major examination,
we have to use what's called 'distributed practice.'
That is, we have to space out our rehearsal of that understanding
in increasingly longer intervals of time ... and the best tool for that is using a diary.
So, what I propose to do now, is to talk you through (for example)
how to learn the quadratic formula
and I'll be talking about those four principles as we do so.
Let's rehearse those four stages, or steps, or principles again
while using the example of deriving the quadratic formula from the quadratic equation.
Remember that step number one was to set time aside ...
to actually be focused and deliberate about your learning.
Remove distractions, find a clean sheet of paper,
set five or ten minutes aside in a quiet place,
and write this at the top of the page.
Now, of course, the quadratic equation is y = ax^2 + bx + c
for any quadratic equation, or any parabola that we're drawing.
But, to find the roots, we solve y = 0 which, in fact, is the x-axis.
So, we're finding the position(s) where a parabola will cross the x-axis.
That's what the quadratic formula is for.
It is, in fact, nothing more than a rearrangement of this equation,
making it very explicit what the x-values are.
Now, we're going to get from there to the formula by completing the square
and we normally don't like a coefficient other than one in front of the x^2.
So, we'll divide everything by 'a' to get rid of this 'a' ...
b/a times x plus c/a = 0/a (which of course is still zero).
Now, we don't know what this value is and, in order to complete this square nicely,
we'll remove this to the other side of the equation.
I'm going to put it over here ... and now we're going to complete the square.
To complete this square we halve the coefficient of 'x'
which is not b/a but b/2a and square it.
And, of course, if we add it on this side of the equation
we have to add it on the other side.
Now that equation is still balanced.
This, of course, since we've completed the square, is (x + b/2a)^2.
There's our x^2 ... there's our (b/2a)^2 ...
and the middle term would be double the product (twice the product).
The product of course is bx/2a, and double it is bx/a.
On this side, if I expand this I have b^2/4a^2.
This term, c/a ...
if I wish to add this to make it into one complete fraction
I must make the denominators the same.
We cannot add or subtract fractions unless the denominators match.
So, to make this match that, I have to multiply it by 4a
to get 4a^2 which means I have to multiply the top by 4a as well.
And that gives me (b^2 -4ac)/4a^2.
Now you can see the quadratic formula starting to take shape.
I'll just copy this again.
Now we take the square root of both sides.
Strictly speaking,
the square root of (x + b/2a)^2 is the absolute value of (x + b/2a).
But I'm going to do two steps at once and remove the absolute value
and replace it with a +/- sign on this side.
The square root of that is this.
The square root of this is the square root of (b^2 - 4ac)
and the square root of 4a^2 is 2a.
We want 'x' by itself, so we remove this term from this side by subtracting it,
which means we subtract it from this side as well.
I'm going to write it at the front ... +/- ...
and, because these do have the same coefficient (denominator),
I can put them into the one expression.
So, it's x = (-b +/-sqrt(b^2 - 4ac))/2a.
And, there it is!
Now, it's a little cramped because I wanted to fit it on the screen.
It's taken one, two, three, four, five, six, seven lines of work ... which is not excessive.
That's fulfilled our second requirement. We've started with something we knew.
We've used a skill that we should understand ... and,
by the way, this will help you practice it (that is, completing the square) ...
and, using our basic algebra, we've ended up with the quadratic formula.
Step number three is to use what we call 'massed practice.'
That is, this I would recommend doing at least once per day.
If not, two, three, four times a day ... whatever you feel you're able to do ...
and, as well as that, to use this formula to solve quadratic equations.
The more of it you can do over a period of a few days or the first week or so, the better.
This is our massed practice.
It establishes the concept and the understanding in our mind.
The fourth step, once we're confident that we understand [the work],
is to set up a system for 'distributed practice.'
That is, for rehearing this in (perhaps) a couple of months time
and then a few months after that and a few months after that.
That requires the use of your diary ... and what do I recommend?
I recommend you ask those three questions that I've mentioned in another video:
Do I understand?
And, hopefully, after your massed practice of doing this derivation
and practicing the use of the equation, you do.
The second question you ask is, "Am I fast enough?"
And again, hopefully, if you've practised enough, you will be.
Of course, if your answer is "No" to either of these questions,
you'll need to do more massed practice.
You'll need to set more time aside in that week, or week-and-a-half,
to do more examples, to do more derivation practice,
so that you're understanding and your speed will improve.
Once you're confident that you've established those,
your third question is,
"How long before I forget" or, "How long before I'm unlikely to remember?"
Now, it might sound funny, but we are very, very good at forgetting.
I know of cases where people have been marooned on islands in the Pacific,
or have been shipwrecked and helped to survive by native populations
where the people did not speak the mother tongue of the sailor.
Mostly, they're English stories that I'm aware of
(but there are some Spanish and others).
And after a few years, they forget their native language.
It's quite 'staggering.'
Something that you would think you would never, ever forget!
And, certainly, it's easy to forget THIS.
So, even though you might understand and you might be fast enough,
if you want to remember it for an exam in six month's time,
you must ask yourself, "Am I confident of remembering it in two or three months?"
Try to judge that time carefully and put a note in your diary
(to recall) to practice this derivation and do a few more questions BEFORE you forget.
In most cases it will be a matter of a month or two to start with
and, hopefully, the next time you go through these questions
it'll be 3-4 months, and then 6 months
until this becomes as well established as, perhaps, Pythagoras' Theorem
or even just adding and subtracting fractions or something like that
(that you should be able to do very, very well).
But, there it, is ... those four principles:
set time aside and be deliberate,
start with something you know and 'link it in'
(of course, if you can do this with song as well, or with a story, that's even better),
do your massed practice,
set up your distributed practice with a diary,
and you should have no trouble at all remembering your formulae.
So there we are!
I hope this video has been profitable for you.
Please leave a comment. Please 'like' the video.
And, of course, I'd be delighted if you subscribed to this channel as well.
Thank you very much for watching!