- Hi there, welcome to Monash Open Day.
I'm gonna tell you about how,
what and why I think about maths.
My name is Norm Do and
I'm a senior lecturer
in the School of Mathematics
at Monash University.
Part one of my talk is
where I tell you about
the only topic on which I'm
the world's foremost expert.
What do you think that might be?
Now it's not gonna be
some part of mathematics,
but actually it's gonna be about me.
Pretty much the only topic
I can really sort of claim
to be an expert on.
Now, the reason why I wanna tell you this
is because I think if
anyone tells you anything,
you need to know who they are.
For instance, if a head
of state, for instance,
with no credentials in the health sciences
decides to tell you about
their cure for COVID-19,
maybe don't believe them as much
as your trained medical practitioner.
Anyway, my story begins as
most people's stories do,
when I was born.
And I'd like to tell you as a child
that I quite enjoyed mathematics.
Now, before you think
that's a little bit strange,
I'm here to claim that actually
this is just the natural state for humans,
to love mathematics.
If I think for instance
about young children,
in particular my own
six-year-old and three-year-old,
and what they love to do.
I see that they love numbers.
They love playing with ideas.
They love playing with shapes
or creating things with Lego and blocks.
They love asking questions about the world
and discovering new things.
And I think this is the true
spirit of what mathematics is.
And somewhere along the way during school,
a lot of people for one reason or another
somehow lose that love of mathematics.
So I'm here to remind you
exactly what maths is about.
Now that was me as a child.
I'd like to fast forward to me
when I was finishing school
'cause many of you are
at that stage right now.
And I'd like to tell
you about how I ended up
choosing my university courses.
You see, I progressed through
school enjoying mathematics
most of the way.
And I knew when I finished school
that I still loved mathematics.
So I thought I'd do a science degree
where I could really satisfy my passion
to discover more about maths.
But I thought I'd do
something else as well.
And I was quite interested in
the arts and the humanities.
So being passionate
about these two things,
I thought a combined degree
in the arts and sciences
would suit me well.
So is that what I ended up studying?
As it turns out no, because
this is what happened.
On the night before my
course preferences were due,
I had a very concerned
mother of a friend of mine
called me up and
desperately pleaded with me
not to study mathematics.
She said, don't do this, Norm.
You're throwing your life away.
You'll come out of the end of your degree
and not be able to get a job.
Or maybe you'll be able to get a job
but it'll be something like
a high school mathematics teacher.
Now how wrong she was, and
I didn't know at the time
'cause I was young and naive,
but I do know now doing
a degree in mathematics
places you very well for a career.
And if that career happens to be
in high school mathematics teaching,
then I think that's a
particularly noble thing to do.
And some of the happiest people I know
are high school mathematics teachers.
Anyway, she said to me,
if you're really keen on doing maths,
at least to an engineering
degree on the side.
Forget about the arts.
And silly me it turns out
that I listened to her.
So actually one of my pet peeves now
is to hear students say
things like I like math,
so I'm gonna do engineering.
What I think people should say
is if they love engineering,
they should do engineering.
But if they love maths,
maybe think about studying mathematics.
So I persisted with my engineering degree.
It's not something that I use today.
As I said, I'm now a senior lecturer
in the school of mathematics.
Now let's fast forward to my job now.
What do you think I do
as a senior lecturer
at the university?
You might think I spend
all my days lecturing,
and actually that's not quite the truth.
You see, my job entails about one third
teaching and lecturing,
about one third doing
mathematics research,
and about one third checking my emails
because almost any job today
involves a lot of that.
Now I won't tell you about
the teaching or the emails,
but I'll say something about the research
because that's something that most people
don't know too much about.
You see, when I say
meet people at a party,
or if I'm talking to the
person next to me on a plane,
or if I'm talking to the taxi driver
taking me to the airport and
they asked me about what I do,
and I told them that I do math research,
most of them don't know what that means.
If they push me further to ask,
well, what does that entail?
I'd say, well, it really
means that I'm trying
to create new knowledge in mathematics.
I'm trying to solve new
problems in mathematics.
For a lot of people,
their first reaction is,
well, wasn't there enough already?
And I claim not, maths is a
very expensive discipline.
There are still so many
things we don't know,
and it's a real joy and
it's very satisfying
to be part of a community,
which is trying to develop
this new knowledge.
Anyway, maybe that's
enough about me for now.
Let's move on to part two of my talk.
And the next part of my talk,
I'd like to tell you what math is about
and how school maths is only
really the tip of the iceberg.
We're gonna ask the
question, what is maths?
Now, if you wanna know what maths is,
in fact, if you wanna
anything in this world,
what do you do?
Of course, the first thing you do
is you go straight to Wikipedia
and you type in maths.
And what does it tell you?
Well, Wikipedia claims that
maths is about three things.
First studying quantity,
structure, space, and change.
Second, it is about seeking out patterns
and formulating new conjectures.
And third is about resolving conjectures
by mathematical proof.
Now, my claim is that in school,
we don't actually see the
whole diversity of mathematics.
In fact, really out of these three things
we spend most of our time in
school mathematics learning,
only spending time on the first?
Yes, we do study quantity
in the form of numbers,
structure, space, in the form of geometry,
and change, which essentially relates
to things like calculus that
you might be learning about.
But we spend very little time
on seeking out patterns and
formulating new conjectures,
and very little time on
resolving conjectures
by mathematical proof.
So my claim is that the
real essence of mathematics
isn't fully portrayed by our sort of,
by our education at school.
Now, of course, Wikipedia
is all well and good.
But if you want an alternative
but equally reliable source of knowledge,
you could ask Norm, that's me.
So what would I say maths is about?
Well, I'd say it's about
these three things.
First solving concrete problems.
Second, playing around with ideas.
And third analyzing problems
beyond their solution.
And again, I claim that we don't do much
of these things in the school setting.
So I should be clear about
what these things mean.
When I say solving concrete problems,
well, I mean that the problems
have to be in some sense,
a little bit explicit.
If you just said, well,
I wanna solve the problem
of world peace, then possibly maths,
it's not so clear how maths has something
to say about that problem.
That's not to say that it doesn't,
but maybe you'd have to
sort of refine your problem
and look deeper into it
and make it a little bit more concrete
and explicit before you can bring
your mathematical knowledge
to bear on that problem.
When I say solving concrete problems,
I also don't mean solving
problems about concrete.
But that is not to say that maths doesn't
or isn't able to solve problems
you might have with concrete.
In fact, I used to have a friend
who worked at the concrete company
and indeed used mathematics
to solve problems there.
So maths I claim is about
playing around with ideas.
And this is really what I
think the bulk of my job
in research is about.
I sit there and I think and I play around,
and I try and discover new
mathematics in that way.
And this third one is maybe something
that people don't talk about much,
but the spirit or the nature
of being a mathematician
is to analyze problems
beyond their solution.
As soon as you solve one, of course,
you sit there patting
yourself on the back.
But the next thing you
do is you ask, well,
what other problems does this lead to?
The tools that I've used in my solution,
can they be taken to solve a different,
maybe more difficult problem?
So as I'm saying, and I'll say it again,
maths is more than what you see at school.
And you might ask, well, why is that?
And one of the reasons is that
the school maths that we do
is enabling, it's enabling
us to do more in mathematics.
But if you don't go into
study more mathematics,
you never see that.
So people often say that school maths
is like the vocabulary, the spelling,
and the grammar of English.
Now, vocabulary, spelling, and grammar
are of course very important.
They enable you to do useful
things like writing letters
and creative things like writing stories.
School maths is very similar.
It enables you to do
useful things like solving
real world problems
that require mathematics
as well as creative things
like solving interesting,
difficult problems.
The issue though is that
in the school setting,
you don't see much of these
useful and creative things,
which is why, if you
really love mathematics,
I'll encourage you to go on
studying it beyond school.
The other impression that
school maths gives you
is that maths is quite limited, right?
That there are only a few topics.
And that may be once you've
understood those topics
you know a lot of mathematics.
And I'm here to tell you
that that is certainly
not the case.
The world of math is
essentially limitless.
And to really sort of make the point,
if you look at how much new mathematics
is being produced every
year, there's tons of it.
More than 75,000 articles containing
new mathematical results appear each year.
And as I've said before, as
a mathematical researcher,
I'm part of the problem.
One of the other things that you might
think about mathematics is that
there's a sense of linear
progression through mathematics.
And this is what school
somehow tells you, right?
You start off doing year,
like grade one mathematics,
then grade two mathematics,
then grade three.
And maybe if, you know,
if you go all the way up
to year 11 and 12, you
start to realize, well,
maybe there's a little bit
of choice in year 11 and 12.
But mathematics is way
more diverse than that.
If you go on to study mathematics
at the university level
and particularly if you come
to Monash and study with me,
there are subjects like
calculus and linear algebra,
which you learn about in
your first year of studies.
But after that, the number
of topics just explodes.
You can go on to do
things like statistics,
multivariable calculus,
group theory, number theory,
real analysis, probability,
computational mathematics.
The list goes on and on.
Maybe I've sort of cheated
here and suggested that
it's an almost infinite list.
That's not quite the case.
But there are several lines which sort of
just don't fit onto my slides here.
Okay, well, that's the end
of part two of my talk.
Let's go on to part three.
Now I know you probably
didn't come here for this,
but as a math lecturer,
I can't help but try and teach you
a little bit about mathematics.
So in this section of the talk,
I'm going to tell you about some maths,
which hopefully you don't already know,
and I hope you find it amazing.
Now there are thousands
and thousands of things
I could have pulled out for this,
but I had to choose just one.
But hopefully I've chosen something
that appeals to everyone,
something you can do in your own home.
Okay, so what do you need for this?
You need a long strip of paper.
Okay, now don't run off
and get one, it's okay.
We'll just do this mentally.
What we're gonna do is
take a long strip of paper.
And if you don't have it with you,
just do it with your arms.
It looks like this if
you can see there, okay?
And what we're gonna do is try
and fold the strip of
paper in half, right?
Like so.
And then unfold that strip of paper,
but make that fold 90 degrees.
So with your arms, you
should be forming something
like the Y from YMCA.
On my slides you should be able to see,
it looks a little bit
like this, this V shape.
Now, nothing too exciting there,
but let's maybe try and make things
a little bit more complicated.
What if we took that long strip of paper,
fold it in half, then fold in half again?
Okay, and now what we'd
love to do is unfurl
our piece of paper and make
all the folds 90 degrees.
It's a little bit hard to do,
but you should get something
a little bit like this.
Okay, maybe instead of looking at me,
let's see what the computer
says you should get.
Something like this shape.
Again maybe you don't think
this is terribly exciting.
But then let's try it again.
And I don't suggest you
do this with your arms.
If you really wanna do it at home,
pull out a strip of
paper and give it a go.
But let me do this for you.
This is what the next step looks like.
So you can see it's getting
a little bit more intricate,
but possibly not enough for you to think
that this is amazing.
But let's just keep doing it.
I'll do it a whole bunch of times.
And we'll see what happens.
This is what you get if
you could do this 15 times,
and I hope you'll agree.
This is not something that you
probably would have expected
when we started this exercise.
We get this beautiful picture.
And this thing in mathematics is called
the dragon curve or the dragon fractal.
You might ask why that is.
I mean, I guess if you're sort of
are a little bit shortsighted,
but myopic you might actually think
that this looks a bit like a dragon.
So that's why it's given that name.
And it's a really amazing object.
And of course, as you can see,
it came about despite a very simple rule
of folding, folding
again and folding again.
And then unfurling your piece of paper
until all the folds are 90 degrees.
Now, I just wanna point
out some things about this,
which are amazing.
There are many things I could say,
but here are a couple
of little observations.
One, when you do this,
your piece of paper once you unfurl it,
will never try to sort of
like cross over itself.
You know, if it sort of
comes to meet itself,
then it sort of goes back out again.
It never crosses over itself.
That's the first thing.
And here's another little observation.
Let's consider this
like as if it was a map
and maybe that this
dragon picture that we see
is actually an Island.
Now, if you sort of focus in this sort of,
in a sort of section, which is,
I guess you could call like
a sort of back of the dragon,
or if you think of this as an island,
if you look at these bays
that are being formed
all over the place, they're
quite intricate themselves.
And you might recognize that these bays
in this island are
precisely the same shape
as the original island,
as the dragon itself.
What I'm trying to tell you
is that you could take this shape
and actually tile your
bathroom with it, right?
You could take a whole bunch of these
and actually slot beautifully
against each other.
And you could tile a whole bathroom.
Obviously, maybe don't go out
and change your bathroom tiles just yet.
This is sort of a little bit tricky
to actually cut a tile like this.
So I think this little
mathematical exercise
really sort of, I dunno,
sort of tells me a little bit about how
amazing mathematics can be.
It sort of suggests that almost
anything that you look at,
if you have a very mathematical mindset
can turn out to be interesting,
even if it's as simple as
just following and unfolding
a piece of paper.
Some of you might be studying
mathematics right now,
and I hope you also continue
to do it in the future.
So I think it might be
a good time to tell you
about what I think the right
way is to learn mathematics,
to make it easier and more fun.
So let's think about how I learned maths,
but also that it also speaks
to how I teach mathematics.
So let's get one thing
out of the way first.
And that's the admission
that maths is difficult.
It's not just difficult for
you, it's difficult for me.
It's difficult for everybody.
And maths problems can be hard.
As a famous example, Andrew Weil's,
the British mathematician,
spent eight consecutive years
working on the famous problem
called Fermat's Last Theorem
before solving it finally in 1994.
Now, of course he didn't work continuously
without food and without bathing,
but he did think pretty
hard about this problem
before cracking it.
Similarly, I've thought about
problems for a long time.
I've got one at the moment
that I've been working on
for several months and
I haven't sold it yet.
But every time I sit down to give it a go,
I feel like the solution
is just around the corner.
Now, if maths was easy, I
claimed that it would be boring.
The challenge of doing
mathematics brings about
productive struggle and
makes you a better person
to just enjoy that
struggle and that journey,
even if sometimes the
solution seems very far away.
But even though math is difficult,
I claim that you can do it.
I've met many people in
the mathematics world.
I've taught many students
at all possible levels.
And I've never met anyone
I would call a mathematical genius.
So you don't have to worry about
whether you're a genius or not.
You're not, I'm not, no one is.
What you just need is the right attitude,
a patience for learning mathematics,
and a real passion for challenges
to really enjoy that struggle.
And don't be afraid to be wrong.
I must estimate that probably
more than 90% of the time
when I'm doing mathematics research,
I'm probably not writing down something
that eventually leads
to what you might call
the end result in some sense.
But actually a lot of these
wrong parts do help you
get a feel for where the actual
solution lies to a problem.
So I said I was gonna teach you
about how to learn maths.
In fact, I lied.
I wanted to talk about
how to learn Chinese.
So imagine you're given
the task of learning
all of the approximately
20,000 Chinese characters,
a bad way to do it would be
to pull out 20 random ones
and just try and stick them in your head
and commit them to memory.
And then on day two to pull out 20 more
and then just stare at them
until they're stuck in your head
and then continue in the
same vein in day three,
day four, et cetera, because
by date five, 10, 20,
whatever it is,
there's no way you've remembered anything.
So I claim that what's missing
when you're trying to
memorize Chinese characters
or mathematics or indeed
anything that you're learning
is context and story
because the human brain
is built very well for
remembering these things.
So I think we have to remind ourselves
that Chinese was not
invented to torture us
with memorization exercises.
But it was invented for a reason.
The reason being to
communicate in the written form
between human beings.
Also not only was there a reason,
but it was created somewhat organically.
People probably started drawing pictures
to represent objects,
which then sort of morphed
into the characters
that we see to this day.
So if we take a look at this
list of Chinese characters,
you see that maybe people drew
a man as a full stick figure
as we might do now.
But then over time,
it got shortened to maybe just the legs.
This character for big
is the man talking about
the fish that got away.
It was this big.
The character for mouth was
pretty self explanatory.
And the word, the character for say,
you see the tongue wagging in the mouth.
The character for word,
you see the hot air rising from the mouth.
That all makes sense.
And now there's a context and hopefully
it's much easy to remember
not just 20 random characters,
but these characters and the
interconnections between them.
If we look at the
character for tree, well,
it sort of looks like a tree.
More interestingly though is
the character for beginning,
which is the tree with a
little tick mark at the root
or the beginning of the tree.
And of course end is
the tree with tick mark
at the other side.
The character distress I
find pretty interesting
because yes, I would be distressed
if I had a tree in my mouth too.
Finally, let's look at this last line.
Home, we see this frying
pan lid at the top
and underneath this other character,
the frying pan lid is a roof.
And the character underneath
I'm told is a pig.
And so this harks back
to the time when people
kept domestic animals in
or around their homes.
Honor is a man standing by his word.
And that's not to say
that people of all genders
can't have honor, of course,
but this is just the history
and the story as it developed.
And it's come to us
organically in this way.
The character for good has
a mother next to a child.
So it's saying that the relationship
between mother and child is a good one.
Peace is sort of funny.
We see the, again, this frying pan lid
at the top of the roof
and underneath the character for a woman.
And quarrel, the character,
two characters for women,
or two women side by side.
Okay, so I didn't invent
the language, obviously.
This is just the stories that can be used
to memorize or to help you
to learn these characters.
And I think there's a
lot we can say by analogy
about the way that we
should learn mathematics.
So first is to ask why and ask it often.
Where did it come from, right?
And what does that mean?
Why does it even exist?
Also keep in mind that
things are easy to remember
when they make sense.
So remember the idea
behind the mathematics
that you're learning
not just the end result.
And of course finally remember that maths
was developed by humans for a reason.
If you know the reason,
then the maths will
definitely make more sense.
In this final part of my talk,
I'd like to tell you about why
I think more people
should study more maths.
And that certainly
includes you particularly
if you're not sure about what
career you'd like to have yet.
Now, if I told you that I
thought that maths is useful,
I think you'd probably say that you agree.
But I'm pretty sure you don't know
the extent to which maths is useful.
And I'd like to try and
convince you of that now.
So let me start with an example,
which is the maths of
the Google search engine.
So everyone's heard of Google, of course.
And one of the reasons why
the Google search engine
is so successful is that when you search
for whatever you're searching for,
it doesn't just give you a random list
of web pages that
include your search term.
What it does is it ranks these pages
based on something called page rank.
Now you might think it's called page rank
because it ranks pages.
Incidentally, that's not the case.
It's actually called it
because it's named after
one of the cofounders
of Google Larry Page,
the basic premise of page rank
is that you think of these
webpages as somehow dots.
And when one web page links to another,
you think of an arrow pointing
from the first dot to the second.
And what page rank does, it
essentially tries to measure
the popularity of these dots.
So you can think of them
as people if you like.
And maybe a link is just if
someone likes that other person.
Now, of course, to be a popular person,
not only do you have to have
many people who like you,
but you are even more popular
if those people who like
you are popular themselves.
And when you take into this account
and express it in a mathematical way,
what you've done is you've
recreated page rank.
Now, I like this example not
only because it's somehow,
because everyone knows about it,
but because we actually
give this to our first year
mathematics students at Monash University.
We give them a very small toy example
of the internet Internet
and ask them to implement on a computer,
the page rank algorithm.
Now I wanna speak about careers
and I could tell you about
all the opportunities
available to you if you study mathematics.
But you shouldn't believe me.
Instead, believe the Internet.
Go to this website, careercast.com,
and see what they've written there.
You see last year in 2019,
they took 220 jobs and ranked them
from best to worst based on four factors.
The first being the income
that you make in that job,
the second being the growth
outlook of that sector,
so how likely you are to get
a job now and in the future,
the work environment,
and finally the stress
of being in that job.
Now, if you're like me and you
know that this list exists,
you go straight to the website
and you jump to the bottom.
You wanna know what not to be.
And so here are the six worst
ranking jobs in the list.
But what I'm here to do today
is to show you what the top 10 looks like.
And I claim that most of the jobs
in the top 10 require fundamentally
mathematical knowledge.
So let's start at number
10 which is actuary.
If you don't know what an actuary is,
it's essentially someone
who uses mathematics
and statistics to measure risk.
And they often work for
insurance companies.
Number nine is an
operations research analyst.
So you may not know
but operations research
is a particular branch of
mathematics dealing with
essentially optimizing operations
and processes and so forth.
Then we come to number eight,
which is mathematician,
someone who uses mathematics
fundamentally in their job
every day of course.
Number seven is an
information security analyst.
So as you can imagine,
information security is a very big sector
and it's gonna be an increasingly
big sector going forward.
It combined skills
coming from mathematics,
statistics and computer science.
Then we'll skip a few
and go to number two,
which is statistician.
Obviously someone who uses
statistical methods in their job.
And number one is data scientist.
And I'm sure many of you know that data
is a sort of very big, you know,
the data science industry
is a very big sector
at the moment and ever-growing.
And it requires skills
again from mathematics,
from statistics, from computer science
and so forth to deal with large data sets.
So, the Internet tells you
that if you study mathematics,
that's gonna set you up
for a pretty decent job.
But still I find selling math careers
to students pretty difficult.
And I think one of the reasons is because
maths is a skill rather than a profession.
So people know that if
you study a law degree,
you'll end up as a lawyer.
Or if you study an engineering degree,
you'll end up as an engineer probably.
But math is a bit different to that.
Because it's a skill,
there are all sorts of
different jobs you could end
up in if you study mathematics.
And I find that this is
actually one of the advantages
to studying something like maths.
Because if you go and look at
all the sort of jobs available to you now,
you probably haven't heard
of the vast majority of them, right?
And so it's not clear what sort of studies
or what sort of degree will set you up
to work in that profession.
And indeed math being a skilled
means it's widely
transferrable and applicable.
And that's going to be
increasingly useful over time
as we find maybe in 10 years from now
a lot of jobs that don't
even exist right now.
So here's just a little selection of,
I could've listed probably about 100 jobs,
most of which you haven't heard of
and all of which fundamentally
require mathematical thinking.
So I hope you're at least
a little bit convinced
that mathematics is a good thing to do.
And if that's the case,
I'd love to see you at Monash University
where you can study under
lecturers like myself.
But in the meantime, good
luck with everything.
And I hope to see you in the future.
