Jane: This is But Why: A Podcast for Curious Kids from Vermont
Public. I'm Jane Lindholm. On this show, we take questions
from curious kids just like you, and we find answers. I had a
math teacher for a parent, so for as long as I can remember,
Math has been a part of my life. My stepdad made me and my
brother do math games on long car rides, and he still gives us
math riddles whenever we go visit him now, even though we're
adults. He wishes us a happy Pi Day every March 14, and
suggested we use the Fibonacci sequence for our essential
passcode numbers. Don't know what the Fibonacci sequence is?
You will by the end of this episode. Math and numbers are
fascinating, but I also know it can sometimes get difficult to
wrap your head around big math concepts in school or on long
car rides with a math-loving adult. So today, we're going to
demystify math. You've sent us a lot of mathematical questions,
and our guest today loves all of them. Dr. Melania Alvarez is the
Outreach Coordinator for the Department of Mathematics at the
University of British Columbia in Canada, and she's the
Education Coordinator for the Pacific Institute for the
Mathematical Sciences.
Dr. Melania Alvarez: So my job is to go all around, showing
people how wonderful and interesting and exciting
mathematics really is. If you invite me to your school, I go
to your school and bring some interesting games and puzzles
and things like that to show you, you know, how much fun you
can have doing mathematics.
Jane: Can you give me a puzzle?
Dr. Melania Alvarez: Oh, there are many, but one, one that I
have. So there's a farmer, and he sells chickens. So he goes to
the market, and in the he's going to go to three markets.
And so there goes to the first market, and he sells half of his
chickens plus half a chicken, okay?
Jane: Okay
Dr. Melania Alvarez: And then he goes to the second market and he
sells half of his chickens plus half of a chicken. And then he
goes to the third market and sells half of the chickens that
are left, plus half a chicken. And that's it. And then he has
zero chickens, right? That's it. After that.
Jane: He sold all of his chickens.
Dr. Melania Alvarez: He sold all of his chickens, so yeah, no
chickens left. So how many chickens did he have? When he
sold the chickens, they were all alive.
Jane: I don't know.
Dr. Melania Alvarez: He sold all live chickens,
Jane: All of his chickens,
Dr. Melania Alvarez: All the chickens were alive. Yes, half a
chicken cannot be alive, right? So how many chickens did he
originally have?
Jane: I see, okay, I feel like I need to write it down on paper,
though. So while, while I'm working on this, let me ask you
some of the questions that kids have sent us about math,
starting with some kids who want to understand, who created math?
Sophie: Hi, my name is Sophie, and I'm eight years old, and
live in Arlington, Virginia. Who invented math?
Oonas: My name is Oonas, and I'm six and a half years old, and I
live in Brooklyn, New York. Who invented math?
Dr. Melania Alvarez: Math is something that grew through
thousands of years. It's like a magical tree that was planted by
many, many people, by the Chinese, the Mayans, you know,
so ancient people needed to, needed to count, at some point,
when populations started to grow and we started to have cities,
we people needed to count. We needed, people looked at how
much property they had. There are some cultures where they
only have one, two and many. But you know, the more you have, you
need to, you know, to count. You need to measure. You need to
trade. So we started inventing numbers. Humans started to
invent numbers and systems and symbols and rules and but here
is the twist. So we started to invent that. But at the same
time, math is not just invented, it's also discovered. There are
also things there that are there and we and we discover, like a
triangle has three sides. We didn't, you know, invented that.
You know, it's that's something that is true. So there are
things that we created in order to make sense of a lot of
things, and with that invention, we also discover a lot of
things. So who invented math is we humans started to observe,
started to look for patterns out of need, and we created systems
that allowed us to describe what we were seeing, and that's what
math is.
Jane: So Leilei wants to know who created math problems, and
if we needed to know how to do math as humans, especially as
you say, as we started to have communities and use money or
barter for things. But maybe Leilei is asking, morelike, who
created the study of math, or this idea that we teach people
math and do math problems, not just learn how to count, because
we have to know how to count.
Dr. Melania Alvarez: Well, what happened is that the moment we
are faced with a situation like, okay, I need to get to school,
but first I need to have breakfast, I need to wake up, I
need to brush my teeth. So then you start thinking, okay, how
much time I'm going to take for breakfast, and how much time do
I need to get from here to school? So natural events that
happen to us, and these are problems that we solve in order
to be able to deal with some realities in life. So who
created problems? Well, yes, sometimes you have your teacher
creating tons of problems so that you suffer through them and
stuff like that. But math problems come from the world.
When we start to wonder, you know, the Egyptians, it's like,
Okay, how do I build this pyramid so that it doesn't fall
down, or it doesn't crumble, or a bridge as well, an engineer,
I'm going to build this bridge. How do I build this bridge so
that it will resist trucks going on top? Or how, if this young
girl likes chocolate and she has to divide it equally with
siblings, how do I divide it in a way that is fair for
everybody, you know? Or I strategize perhaps. How do I do
it so that I can get more chocolate than others. So this
is where math, where really math problems come from. Now that
sometimes we create ridiculous problems, like my mother bought
300 watermelons and divided between me and my siblings.
Well, those are weird problems that sometimes I made for
school. But real math problems really come from the world.
Jane: Leilei also wonders, why does one plus one equal two and
not 11?
Dr. Melania Alvarez: We have this system of writing numbers
that is called place value. So it depends where the number is.
So one one means that the one to your right, the first one, is a
one, but the next one is not the one. It's a 10, you see. So for
example, the number 245: in that number, the two is a 200, the
four is a 40, and the five is five. So it depends where you
position the numbers that it changes. So one plus one is two
and one one does not represent two, that one represents a 10
and then a one, so 10 plus one is 11.
Jane: Yeah, it's just interesting to think about.
We've created these systems that are supposed to make it easier
for us and easier for us to do these kinds of problems with
other people, so that your brain and my brain can match up, and
we both understand what one plus one equals. I think I see where
Leilei is coming from in that it can be really interesting. It
can kind of blow your mind when you start to think about like,
oh, we had to make this work for all of our brains.
Dr. Melania Alvarez: For everybody, exactly.
Kai: My name is Kai. I live in California, and I'm four years
old. Why is there so many numbers in math?
Dr. Melania Alvarez: Oh, what happened is, as I said, when
counting started, we had very few numbers. You know, it was 1,
2, 3, many, and then somebody, instead of having three sheep,
now tomorrow, they have four sheep or five sheep, and then
they have to add that. So we needed a system that the more we
have, the more needed to be accounted for. And we have a
very, very nice system where, you know, we just basically can
go on forever and ever and ever and ever and ever.
Jane: Well, some of the kids want to know why.
Lily: My name is Lily. I'm nine years old, and I live in
Portland, Oregon. Why don't numbers end?
Isla: Hi, I'm Isla. I live in Clearwater, and why do numbers
numbers stop?
Nicholas: My name is Nicholas. I'm five years old. I live in
Washington, DC. Why do numbers go off forever and ever.
Jane: Why don't numbers end? Why do we never stop adding numbers?
Is there really infinity numbers?
Dr. Melania Alvarez: Yes, yes. You can give me a number and I
can add one to that number, then now you have more numbers. You
know, for example, there's the googol which has a one and 100
zeros afterwards, right?
Jane: That's right before we all knew what the internet was,
googol was just a number. That's googol spelled G, O, O, G O L.
Google, the internet search engine is G, O, O, G, L, E.
Googol, the number, as Melania said, is a one followed by 100
zeros.
Dr. Melania Alvarez: It's a huge number. And then there's the
Googolplex, which is even larger than that. And you say, so
there's a Googolplex, and I can add one to that, and it's bigger
than the Googolplex. And then I can add one to that one. And I
can keep going on and on, adding 1, 1, 1, 1, 1, and 1. And I just
go on, you see. But also infinity can be kind of also
small. Like, for example, I have a mile to go, okay? And so then
I can go half a mile, and then I go half of half of the mile, and
then I get half and half and half of that, and that, do I
ever get there? No, never. And I can go infinitely, going and the
steps are smaller and smaller and smaller. Eventually, in real
life we get there, but mathematically, we just can be
stuck there, trying to get there. But if you always do
half, half, half, half, half, we will not get there. This is what
is amazing about mathematics, is that you can go on with your
imagination, and math is a great companion for that.
Jane: Yeah.
Dr. Melania Alvarez: And it shows you ways that you never
thought that were possible. And that's what is so wonderful
about it.
Jane: It can kind of make your brain hurt when you try to
think, think, you know, all the way out on those things, but
hurt in a good way. I mean, it's just cool.
Dr. Melania Alvarez: So it is true. You talk about the hurt
and about the struggle and all of that, but what happens when
you are able to solve a problem that is really, really hard? How
do you feel about that?
Jane: Really good. Proud of myself.
Dr. Melania Alvarez: Really good.
Jane: Yeah. Like, I figured it out.
Dr. Melania Alvarez: You get this big high, isn't it? You get
like, wow, I'm really smart. I'm at the top of the world. You
know, that's what we mathematicians live for. Do you
know we were working, we struggle, and then when you
finally solve it, and not just solving it, is the beauty
sometimes of those solutions that are just like, how can this
be so perfect? And that's why. And there are mathematicians
that spend a year, two years, eight years, you know, 10 years,
trying to solve a problem, but they think that it's worth it.
So that's another thing. You don't need to be fast to be a
good mathematician. Many kids think that they have to be
really fast problem solvers to be really good mathematicians.
That's not true. You know, you can take your time.
Tryphon: Hi, my name is Tryphon. I'm nine years old, and I live
in Winnipeg, Manitoba, Canada. Why do people hate math so much?
Jane: Why do people hate math so much? And we should say,
certainly not everybody, but math has a reputation, and
people sometimes seem to feel comfortable saying, Oh, I'm not
good at math, or I hate math, in ways that you wouldn't say about
other things.
Dr. Melania Alvarez: That's right. First of all, many times,
is the way that math is presented to us. It's just like
a series of rules, and you're to solve this problem, and that's
it. And that's not math, actually. That's just something
that we call practice. Real math is not that you need to know the
tools. You need to know how to add and subtract and all of
that, yes, but it's like you need a hammer and you need nails
to build a house. So that's what that is. But math is thinking.
Math is strategizing. So math is not just the solution. It's the
way to get there and how and your thinking to get there. So
it's, so many times is how math is... if it is just rote
learning and just memorizing all the time, which we have to do
sometimes. But if that's the only thing that we learn and the
only thing that we do, then is really boring. I'm telling you,
if that was math, there wouldn't be professional mathematicians,
because no, come on, we're not that boring people. We really
like, we're explorers. That's what it is, math is about
exploration. It's about questioning. It's about doing
all that. If you start to dislike math, and this is also
something that happens, is perhaps along the way, and I
find that many people say I like mathematics, but at some point I
start disliking it, because I got lost. So what I tell kids is
this: if you are in class and you understand that everything
and all of a sudden you don't understand something, go and ask
your teacher, but soon. Don't wait a month or two months
afterwards, go immediately and ask your teacher, Hey, I didn't
get this. Could you explain it to me again? Because math builds
on top. You know, it's something that you it's a structure where
you... So if you miss the first floor, then the rest of the
floors are going to be all wobbly and you're not going to
understand. So don't be embarrassed about asking, and if
you're embarrassed in front of the class, then ask after class.
But don't leave something that you don't understand and say,
well later on, I'll catch up. No, no, no, no. So don't get
lost. Go for help as soon as possible.
Jane: That's always good practice. If you don't
understand something, ask a question, get some help. Math is
complicated, and if one explanation of a math concept
isn't working for you and your brain, you can always say, Could
you describe it to me a different way? Or maybe you need
someone to write it out for you, or help you work backwards from
the answer back to the original question. You can ask for
different kinds of help to make sure you understand the math
you're working on in school or at home.
Coming up, why are the numbers in the order they are, and is
there really math in everything? And of course, I still have to
solve that puzzle Melania gave me about the chicken farmer.
Stay with us.
This is But Why: A Podcast for Curious Kids. I'm Jane Lindholm.
Today, we're talking about numbers and math with
mathematician Melania Alvarez. We're going to solve that puzzle
about the chicken farmer in just a little while. But first, some
of your questions about how numbers are ordered and how they
help us count.
Kai: I'm Kai, and I'm six years old, and I live in Saint Paul,
Minnesota. Why is the numbers in order? Why are they in 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, why can't they just be like in different
orders?
Dr. Melania Alvarez: Well, so numbers are like a step ladder.
We make one number, and then we go to the next, and when we go
to the next, and then we go to the next. So, so that's why it
keeps an order, and that order helps us to count. If the
numbers were all over the place, imagine we have here one, and
then comes 25 and they call 36 and how can we count like that?
We can't! So it is like a ladder. We go step by step by
step, and we gave them numbers those names, right? Because we
need them to track things. We need them to trade. We need them
for things in real life. And if we didn't have that order, we
wouldn't be able to do that. We'll be lost in confusion in
the jungle of numbers. So, so that's why there's a rule, why
we have an order. You know, it's like why don't you build a house
upside down? Because then it wouldn't make sense. So that's
exactly the same thing with this.
Miles: I'm Miles and I'm five years old. I live in Quebec,
Canada. Why are numbers for counting?
Dr. Melania Alvarez: Why are numbers for counting? Well, the
numbers are counting because, actually, that's where math
began. Math began with counting, long, long, long, long time ago,
over 20,000 years ago. So this is something that helped people
to survive.
Jane: And you can think about, let's say, you had to get
through a winter, and you're, you needed to have enough crops
for your whole family to get through the winter. Well, you
have to know how many people are there that I'm trying to feed.
So let's say there are four people in my family. How much do
we eat every day? How many days do we need to get through to get
through the winter before we can grow more food. So you need to
be able to count and add and multiply to know how much food
you need to store for the winter, which will take you to
how much you need to grow in the summer, which will take you back
to how much you need to plant and how many plants you think
the animals are going to eat before you get to have them. So
how much more do you actually need to plant to be able to have
enough to harvest... and, uh oh, there's a new baby in the
family. Now you have to do that, but for five people, so you have
to be able to do that math in order to survive, and even if
we're not thinking about growing all of our own food now, as
families, we have to figure out how to budget, how much money we
need, how much money we need if we also want to go on vacation
to the beach at some point. So math is really important, and we
do like it because it helps us get the things that we want and
need, even if we think, or some people think they don't like
math in school.
Dr. Melania Alvarez: You're absolutely right.
Jane: Zoe lives in California and is seven and wonders, why do
you need to do math? We already said math has helped people
survive, but what are some of the other ways math is needed?
Dr. Melania Alvarez: Well, you need to do math for for so many
things. You need to do math to go to the supermarket. You need
to do math to trade with people. You need to do math, to build a
house, to build a pyramid, to study chemistry, to read graphs
about how, who's going to win the presidential election. So we
also need to know, like, for example, if you are in class and
you want to go to recess and you want to know how many minutes
are there to recess? So you can count them. Oh, it's almost
almost five minutes to recess, I want to go there. Also, if you,
if you like, to trade cards. How many cards should I trade for
this card? That card is worth three of these cards. And now,
for example, all this technology, the phones, the TVs,
everything, all of that was created thanks to math.
Saanvi: My name is Saanvi. I'm 10 years old, and I live in
Bothell, Washington. Is there math in everything?
Dr. Melania Alvarez: Math is not just in everything. Math is also
a way to see everything. We can see it in patterns from nature,
flowers. There's a sequence called the Fibonacci sequence
that goes one one, and then what is one plus one, is two. And
then one plus two is three. And then if I add two and three, so
I take the last two numbers and add them up, and then I get the
next sequence. And what's interesting is that that
sequence represents a lot of things that happen in nature,
the petals in flowers, they usually, they are Fibonacci
number. So the really math is like this invisible thread that
connects everything around you, and it is the spiral in the
shell. We have the rhythm in music, when you bounce a
basketball, too. The trees in the branches. But the secret
about this is you can see this if you are curious. You have to
be curious, okay, to see it. So math doesn't necessarily scream
for attention. It whispers to you, says, hey, hey, come check
it out. Well, you have to be curious. You can notice a lot
more if you're curious, and you can have a great time if you're curious.
Jane: Well, you came to the right place, because anybody
who's listening to this podcast is a curious kid. It's in the
name of our show. So all of the kids who are listening, that's
something that we all have in common all around the world. We
are curious kids who want to know more about the world. So if
you have us convinced. Let's wrap up with Alistar's question.
Alistar: Hi, my name's Alistar. How do you become a
mathematician?
Jane: How do you become a mathematician? We all want to do
it now, Melania, how do we become mathematicians?
Dr. Melania Alvarez: So first of all, we are all mathematicians.
You know babies, they immediately can recognize
patterns. When we are playing and we're putting cubes on top
of each other and trying to measure if it's going to fall or
not fall, you know, and how to build things when we're little,
we are all mathematicians, all of us. When we like to say, How
many cookies I need to bake for everybody in my family, I look
at cool pattern, I say, Oh, how does that grow? How can I
continue this pattern, all of that. Now, mathematicians, a
real mathematician, not just answer the questions, they also
ask questions. A mathematician is always asking questions about
how things work, and then tries to answer it. So this is what it
is. Everybody, you know, everybody can do math, and we
all can be mathematicians. Now, if what you want... asking is,
how I become a professional mathematician, usually, most
mathematicians, I mean, there are some who don't, but most of
them go to university, and they... they have become
undergraduates in math or a math related thing, and then they go
for a PhD in mathematics, and they solve problems. Now there
are two types of mathematicians, mainly. The applied
mathematician and the pure mathematician. So I am an
applied mathematician and my husband is a pure mathematician.
So an applied mathematician people come with problems to me,
like I have worked with problems in anthropology. I have worked
with problems in engineering. So engineers come with a problem
and they want, you know, they want someone who does all the
calculations and who creates a model of what they're seeing. So
I do that. So people comes with their problems, and I take care
of their math problems, only their math problems. So that's
right. The pure mathematicians, they ask questions if some
really abstract thing can work or not work, you know, and then
they start, they start working on the on this abstract
mathematics that it looks like they are useless, that who
cares, doing this math, that there doesn't seem to be real
Dr. Melania Alvarez: did he originally have?
application. And lo and behold, 10 years from now, 200 years
from now, 3000 years from now, all of the sudden we find the
application. Isn't it.. it's like magic, and that's what is
so exciting about math.
Jane: So I want anybody who's listening now who wants to solve
it to pause the episode and see if you can figure it out.
Can we go through the answer and how one way that we might be
able to solve it, because I was thinking about it while you were
talking, and I think I have an answer.
Dr. Melania Alvarez: Okay, why don't you tell me what you're
thinking about?
Jane: Okay, I think he had seven chickens at the start.
Dr. Melania Alvarez: And how did you solve that?
Jane: When you said all the chickens were alive and at the
end he had no chickens, it made me realize that when he at the
very end at the last market, he sold half of his chickens plus
half a chicken, he would have to have one chicken to sell,
because no chickens were harmed.
Dr. Melania Alvarez: Yes.
Jane: So then I sort of went from the very end to the
beginning by sort of adding and multiplying rather than
subtracting and dividing. And so I think he had one chicken at
the beginning,
Dr. Melania Alvarez: Yes.
Jane: Three chickens in the middle and seven chickens at the
start.
Dr. Melania Alvarez: And you got it absolutely correct.
Jane: Wow, phew.
Dr. Melania Alvarez: And that's actually, that's the way,
working backwards, you see. So you work backwards. You said,
Okay, I think that the first one is one, and then the next one
will be three, because I will be, calculating what's half, you
know, double of that, and then you calculate what will be sold,
and then that. So you got absolutely right. So in the
first market, he has seven chickens, and half of those
chickens is three and a half plus half a chicken is four
chickens. So seven minus four is three. So we're left with three
chickens. He goes to the second market, half of those three
chickens is one and a half chickens. And then plus half a
chicken, that's two chickens. Three minus two is one. So we're
left with one chicken. And the last market is one chicken, and
then it's half of a chicken is half, and then the other half is
one. And there we are, and oh, there were no chickens hurt in
this puzzle.
Jane: I was so, as you said, I was so pleased when I realized,
oh, I can figure this out and get to the answer.
Dr. Melania Alvarez: So there are many ways that you can solve
a math problem. You can you can guess, and you can do a
systematic guess, not just random numbers, but little by
little, guesses that will give you some information. So you go
on. You can start backwards. You can draw a diagram, too.
Sometimes you can draw a picture, and that will help you.
There are many ways, and that's what mathematicians use. They
have the different ways of approaching and see what work,
what will work, and sometimes it works, and if it doesn't work,
then you start again. And that's all there is.
Jane: Let's end this episode there. Did you figure out the
answer to the puzzle? I will admit it took me a while, and I
had to write myself some notes, and I did start to get really
worried at one point that I wasn't going to get the answer
right. And then my mind started to go blank, and then I thought
of all of you listening, and I thought, oh no, you'll all be so
disappointed in me, or think I'm not very smart. But as Melania
was talking, I realized two things. The first was that I
just needed to take a deep breath and start working
backwards, and I could probably figure it out. And number two,
that if I couldn't figure it out, that's okay. No one is
supposed to just automatically know all the answers to things.
And when we're learning math, part of what we're doing is
learning how to solve problems. So I knew I could ask Melania
for help if I got stuck, and she would steer me to the right
strategy. Thanks to Dr. Melania Alvarez of the University of
British Columbia and the Pacific Institute for the Mathematical
Sciences for answering all of our math questions today. We're
going to add some math puzzles and resources in the show notes
if you want more. As always, if you have a question about
anything, have an adult record you asking it on a smartphone
using an app like voice memos, then have your adult email the
file to questions@butwykids.org But Why is produced by Melody
Bodette, Sarah Baik and me, Jane Lindholm at Vermont Public and
distributed by PRX. Our video producer is Joey Palumbo, and
our theme music is by Luke Reynolds. If you like our show,
please have your adults help you give us a thumbs up or a review
on whatever podcast platform you use, we'll be back in two weeks
with an all new episode. Until then, stay curious.