Conrad Wolfram: Teaching kids real math with computers

Prevoditelj: Frane Šarlija Recezent: Tilen Pigac - EFZG

We've got a real problem with math education right now.

U današnje vrijeme imamo veliki problem s matematičkim obrazovanjem.

Basically, no one's very happy.

U osnovi, nitko nije jako sretan.

Those learning it

Oni koji je uče

think it's disconnected,

misle da je nejasna,

uninteresting and hard.

nezanimljiva i teška.

Those trying to employ them

Oni koji ih pokušavaju zaposliti

think they don't know enough.

misle da ne znaju dovoljno.

Governments realize that it's a big deal for our economies,

Vlade shvaćaju da je to velik problem za našu ekonomiju,

but don't know how to fix it.

ali ne znaju kako to popraviti.

And teachers are also frustrated.

I učitelji su frustrirani.

Yet math is more important to the world

Ipak, matematika je potrebnija svijetu

than at any point in human history.

nego ikad prije u ljudskoj povijesti.

So at one end we've got falling interest

S jedne strane imamo opadanje zanimanja

in education in math,

za učenjem matematike,

and at the other end we've got a more mathematical world,

dok s druge strane imamo sve više matematički svijet,

a more quantitative world than we ever have had.

brojčani svijet, više nego što smo ikad prije imali.

So what's the problem, why has this chasm opened up,

Dakle, gdje je problem, zašto se ovaj jaz stvorio

and what can we do to fix it?

i što možemo napraviti da popravimo stvari?

Well actually, I think the answer

Pa zapravo, ja mislim da odgovor

is staring us right in the face:

leži ravno ispred nas.

Use computers.

Koristite računala.

I believe

Ja vjerujem

that correctly using computers

da je pravilna upotreba računala

is the silver bullet

presudni faktor

for making math education work.

koji će pokrenuti matematičko obrazovanje.

So to explain that,

Kako bi ovo objasnio,

let me first talk a bit about what math looks like in the real world

dopustite mi da prvo kažem nešto o matematici u stvarnom svijetu

and what it looks like in education.

i o tome kako ona izgleda u obrazovanju.

See, in the real world

Vidite, u stvarnom svijetu

math isn't necessarily done by mathematicians.

matematiku ne koriste nužno samo matematičari.

It's done by geologists,

Koriste ju geolozi,

engineers, biologists,

inžinjeri, biolozi,

all sorts of different people --

mnogo različitih ljudi --

modeling and simulation.

za modeliranje i simuliranje.

It's actually very popular.

To je zapravo veoma popularno.

But in education it looks very different --

Ali, u obrazovanju to izgleda mnogo drukčije --

dumbed-down problems, lots of calculating,

glupavi problemi, dosta računanja --

mostly by hand.

većinom ručnog.

Lots of things that seem simple

Mnoge stvari izgledaju jednostavno

and not difficult like in the real world,

i lagano u stvarnom svijetu,

except if you're learning it.

ako o njima ne učite.

And another thing about math:

I još jedna stvar o matematici:

math sometimes looks like math --

matematika nekad izgleda kao matematika

like in this example here --

kao u ovom primjeru ovdje --

and sometimes it doesn't --

ali, nekad ne izgleda.

like "Am I drunk?"

Kao "Jesam li ja pijan?"

And then you get an answer that's quantitative in the modern world.

I tad dobijete odgovor koji je izražen brojkama u stvarnom svijetu.

You wouldn't have expected that a few years back.

Ne biste to očekivali prije nekoliko godina.

But now you can find out all about --

Ali sad možete naći sve o tome --

unfortunately, my weight is a little higher than that, but --

nažalost, moja težina je malo veća od toga, ali --

all about what happens.

sve ovisi o tome što se dogodi.

So let's zoom out a bit and ask,

Hajde da malo pogledamo stvari iz šire perspektive i zapitamo se,

why are we teaching people math?

zašto mi učimo ljude matematiku?

What's the point of teaching people math?

Koja je svrha učiti ljude matematiku?

And in particular, why are we teaching them math in general?

I na kraju krajeva, koja je svrha učenja matematike uopće?

Why is it such an important part of education

Zašto je to tako bitan dio obrazovanja

as a sort of compulsory subject?

kao vrsta obaveznog predmeta?

Well, I think there are about three reasons:

Pa, ja mislim da postoje tri razloga:

technical jobs

tehnički poslovi

so critical to the development of our economies,

važni za razvoj naše ekonomije,

what I call "everyday living" --

ono što ja nazivam svakodnevnim životom.

to function in the world today,

Kako bi funkcionirali u današnjem svijetu,

you've got to be pretty quantitative,

moraš biti poprilično dobar u računanju,

much more so than a few years ago:

mnogo više nego prije nekoliko godina.

figure out your mortgages,

Izračunati svoju hipoteku,

being skeptical of government statistics, those kinds of things --

ako nemate povjerenja u državne statistike, i druge slične stvari.

and thirdly, what I would call something like

I treće, ono što bi ja nazvao kao

logical mind training, logical thinking.

logičke vježbe za mozak, logičko razmišljanje.

Over the years

Kroz godine

we've put so much in society

puno smo uložili u društvo

into being able to process and think logically. It's part of human society.

koje može razmišljati i raditi logički; to je dio našeg ljudskog društva.

It's very important to learn that

Jako je važno naučiti to.

math is a great way to do that.

Matematika je sjajan način za postići to.

So let's ask another question.

Idemo se zapitati još jedno pitanje.

What is math?

Što je matematika?

What do we mean when we say we're doing math,

Što mi mislimo kad kažemo da radimo matematiku,

or educating people to do math?

ili učimo ljude matematici?

Well, I think it's about four steps, roughly speaking,

Mislim da se radi o četiri koraka, grubo govoreći,

starting with posing the right question.

počinjemo s postavljanjem pravog pitanja.

What is it that we want to ask? What is it we're trying to find out here?

Što je to što mi želimo pitati? Što je to što mi želimo pronaći tu?

And this is the thing most screwed up in the outside world,

I ovo je najzeznutija stvar u stvarnom svijetu,

beyond virtually any other part of doing math.

iznad svih drugih stvari korištenja matematike.

People ask the wrong question,

Ljudi pitaju krivo pitanje,

and surprisingly enough, they get the wrong answer,

i začudo, dobivaju krivi odgovor,

for that reason, if not for others.

i zbog tog razloga, ako ne zbog drugih.

So the next thing is take that problem

Dakle, sljedeća stvar je, uzmi taj problem

and turn it from a real world problem

i pretvori ga iz problema iz stvarnog svijeta

into a math problem.

u matematički problem.

That's stage two.

To je druga faza.

Once you've done that, then there's the computation step.

Jednom kad to napravite, onda slijedi korak proračuna.

Turn it from that into some answer

Pretvorite to iz toga u neki odgovor

in a mathematical form.

u matematičkom obliku.

And of course, math is very powerful at doing that.

I naravno, matematika je veoma jaka u tome.

And then finally, turn it back to the real world.

I tada napokon, ponovno vratite u stvarni svijet.

Did it answer the question?

Je li odgovorilo na pitanje?

And also verify it -- crucial step.

I također provjerite rješenje -- to je najvažniji korak.

Now here's the crazy thing right now.

Sad tu imamo ludu stvar.

In math education,

U matematičkoj naobrazbi,

we're spending about perhaps 80 percent of the time

mi trošimo oko otprilike 80 posto vremena

teaching people to do step three by hand.

učeći ljude da naprave korak tri na ruke.

Yet, that's the one step computers can do

Ipak, to je jedan korak kojeg računala mogu napraviti

better than any human after years of practice.

bolje nego bilo koji čovijek nakon godina vježbe.

Instead, we ought to be using computers

Umjesto toga, trebali bi koristiti računala

to do step three

da rade treći korak

and using the students to spend much more effort

i zahtijevati od studenata da troše puno više truda

on learning how to do steps one, two and four --

učeći ljude kako da rade prvi, drugi i četvrti korak --

conceptualizing problems, applying them,

zamišljajući problem, njihva primjena,

getting the teacher to run them through how to do that.

koristeći učitelje da ih vode kroz postupak kako bi trebalo raditi.

See, crucial point here:

Vidite, ključan problem je tu:

math is not equal to calculating.

matematika nije isto sto i računanje.

Math is a much broader subject than calculating.

Matematika je puno širi predmet od računanja.

Now it's understandable that this has all got intertwined

Sad je razumljivo da je sve ovo isprepleteno

over hundreds of years.

kroz stotine godina.

There was only one way to do calculating and that was by hand.

Postojao je samo jedan put da se nešto izračuna i to je bilo na ruke.

But in the last few decades

Ali u posljednjih nekoliko desetljeća

that has totally changed.

to se potpuno promijenilo.

We've had the biggest transformation of any ancient subject

Imali smo najveću promijenu od svih drevnih predmeta

that I could ever imagine with computers.

koju sam ja mogao zamisliti s računalima.

Calculating was typically the limiting step,

Računanje je većinom bilo ograničavajući korak,

and now often it isn't.

a malo kad nije.

So I think in terms of the fact that math

Dakle, ja mislim kako je činjenica da je matematika

has been liberated from calculating.

sada oslobođena računanja.

But that math liberation didn't get into education yet.

Ali to oslobođenje matematike nije još ušlo u obrazovanje.

See, I think of calculating, in a sense,

Vidite, ja mislim o računanju, u jednom smislu,

as the machinery of math.

kao o stroju matematike.

It's the chore.

Ono je sitni posao.

It's the thing you'd like to avoid if you can, like to get a machine to do.

Ono je stvar koju bi htjeli izbjeći, kao što bi htjeli da stroj radi.

It's a means to an end, not an end in itself,

To je sredstvo do cilja, ali ne i sami cilj.

and automation allows us

I automatizacija nam dopušta

to have that machinery.

da imamo taj stroj.

Computers allow us to do that --

Računala nam to dopuštaju.

and this is not a small problem by any means.

I to nije mali problem ni u kom slučaju.

I estimated that, just today, across the world,

Ja sam izračunao, kako danas diljem svijeta,

we spent about 106 average world lifetimes

potrošimo oko 106 prosječnih svjetskih života

teaching people how to calculate by hand.

učeći ljude kako da računaju na ruke.

That's an amazing amount of human endeavor.

To je ogromna količina ljudskog truda.

So we better be damn sure --

Trebali bi biti poprilično sigurni --

and by the way, they didn't even have fun doing it, most of them --

i također, oni nisu imali nikakvu zabavu dok su radili to, bar većina njih.

so we better be damn sure

Pa bolje budimo poprilično sigurni

that we know why we're doing that

da znamo zašto to radimo

and it has a real purpose.

i da to ima stvarno svrhu.

I think we should be assuming computers

Mislim da bi trebali zadužiti računala

for doing the calculating

da rade računanje

and only doing hand calculations where it really makes sense to teach people that.

i raditi ručno računanje samo tamo gdje to stvarno ima smisla učiti ljude.

And I think there are some cases.

I mislim da ima nekih slučajeva

For example: mental arithmetic.

Na primjer: mentalna aritmetika.

I still do a lot of that, mainly for estimating.

Ja i dalje to radim često, većinom za procijene.

People say, "Is such and such true?"

Ljudi kažu, je li to i to točno,

And I'll say, "Hmm, not sure." I'll think about it roughly.

a ja kažem, hmm, nisam siguran. Razmislit ću o tome okvirno.

It's still quicker to do that and more practical.

To je još uvijek brže i praktičnije napraviti.

So I think practicality is one case

Dakle mislim da je praktičnost jedan slučaj

where it's worth teaching people by hand.

gdje je korisno učiti ljude ručnom računanju.

And then there are certain conceptual things

I tu su neke koncepcijske stvari

that can also benefit from hand calculating,

kojima isto može koristiti ručno računanje,

but I think they're relatively small in number.

ali mislim da je njihov broj relativno mali.

One thing I often ask about

Jedna stvar koju se često pitam

is ancient Greek and how this relates.

su drevni Grci i kako je to povezano.

See, the thing we're doing right now

Vidite, ono što mi sad radimo,

is we're forcing people to learn mathematics.

je prisiljavanje ljudi da uče matematiku.

It's a major subject.

To je glavni predmet.

I'm not for one minute suggesting that, if people are interested in hand calculating

Ja ni na jedan tren nisam nagovarao, ako ljudi žele ručno računati

or in following their own interests

ili slijediti njihove osobne interese

in any subject however bizarre --

u bilo kojem području ma kako bizarnom --

they should do that.

onda bi i trebali to raditi.

That's absolutely the right thing,

Potpuno je ispravna stvar,

for people to follow their self-interest.

da ljudi slijede svoje osobne interese.

I was somewhat interested in ancient Greek,

Bio sam zaintersiran za drevne Grke,

but I don't think that we should force the entire population

ali ne mislim da bi trebali prisiliti cijelo stanovništvo

to learn a subject like ancient Greek.

da uče predmet kao drevni Grci.

I don't think it's warranted.

Mislim da to ništa ne jamči.

So I have this distinction between what we're making people do

Tako da ja razlikujem slučaj kad mi želimo da ljudi rade

and the subject that's sort of mainstream

taj predmet masovno

and the subject that, in a sense, people might follow with their own interest

i slučaja kad, u nekom smislu, ljudi slijede svoje osobne interese

and perhaps even be spiked into doing that.

i možda budu navućeni na to.

So what are the issues people bring up with this?

Dakle, koji su problemi koje ljudi imaju s ovim?

Well one of them is, they say, you need to get the basics first.

Jedan od njih je, kažu oni, da trebaš naučiti prvo osnove.

You shouldn't use the machine

Ne bi smjeli koristiti strojeve

until you get the basics of the subject.

dok ne naučimo osnove predmeta.

So my usual question is, what do you mean by "basics?"

Moje uobičajno pitanje je, što mislite pod osnove?

Basics of what?

Osnove čega?

Are the basics of driving a car

Da li je osnova vožnje automobila

learning how to service it, or design it for that matter?

znanje kako ga popraviti, ili dizajnirati za neku namjenu?

Are the basics of writing learning how to sharpen a quill?

Da li je osnova pisanja znanje kako naoštriti olovku?

I don't think so.

Ja ne mislim tako.

I think you need to separate the basics of what you're trying to do

Ja mislim da treba odvojiti osnove onog što želiš napraviti

from how it gets done

od toga kako se to napravi

and the machinery of how it gets done

i postupka kako to napraviti.

and automation allows you to make that separation.

I automatizacija omogućava da to razdvojimo.

A hundred years ago, it's certainly true that to drive a car

Prije sto godina, sigurno je za voziti automobil

you kind of needed to know a lot about the mechanics of the car

trebalo znati dosta o mehanici automobila

and how the ignition timing worked and all sorts of things.

i kako radi tehnika paljenja i niz drugih stvari.

But automation in cars

Ali automatizacija u automobilima

allowed that to separate,

dopustila nam je da to razdvojimo,

so driving is now a quite separate subject, so to speak,

tako da je vožnja sad poprilično odvojena, tako reći,

from engineering of the car

od inžinjerstva u automobilima

or learning how to service it.

ili znanja kako ih popraviti.

So automation allows this separation

Dakle, automatizacija dopušta ovo odvajanje

and also allows -- in the case of driving,

i također dopušta -- u slučaju vožnje,

and I believe also in the future case of maths --

a ja se nadam i u slučaju matematike u budućnosti --

a democratized way of doing that.

demokratičniji način rada.

It can be spread across a much larger number of people

To se može primjeniti na puno veći broj ljudi

who can really work with that.

koji mogu stvarno raditi s tim.

So there's another thing that comes up with basics.

Dakle ima još jedna stvar koja dolazi s osnovama.

People confuse, in my view,

Ljudi miješaju, to je moje viđenje,

the order of the invention of the tools

redoslijed izuma alata

with the order in which they should use them for teaching.

s redoslijedom kojim bi ih trebali koristiti za učenje.

So just because paper was invented before computers,

Samo zato što je papir izumljen prije računala,

it doesn't necessarily mean you get more to the basics of the subject

to ne mora nužno značiti da dobivaš više osnova nekog predmeta

by using paper instead of a computer

ako koristiš papir umjesto računala

to teach mathematics.

za učenje matematike.

My daughter gave me a rather nice anecdote on this.

Moja kćerka mi je dala jako lijepu pričicu o ovoj temi.

She enjoys making what she calls "paper laptops."

Ona uživa u izradi papirnatih laptopa, kako ih ona naziva.

(Laughter)

(Smijeh)

So I asked her one day, "You know, when I was your age,

Tako sam ju ja jednog dana pitao, "Znaš, kad sam ja bio tvojih godina,

I didn't make these.

ja nisam radio to.

Why do you think that was?"

Što misliš zašto?"

And after a second or two, carefully reflecting,

I nakon sekundu ili dvije pažljivog razmišljanja,

she said, "No paper?"

kaže, "Nije bilo papira?"

(Laughter)

(Smijeh)

If you were born after computers and paper,

Ako ste se rodili nakon računala ili papira,

it doesn't really matter which order you're taught with them in,

nije veoma važno kojim redoslijedom ih koristite za učenje,

you just want to have the best tool.

samo želite imati najbolji alat.

So another one that comes up is "Computers dumb math down."

Druga stvar do koje se dođe je kako "računala zatupljuju matematiku."

That somehow, if you use a computer,

Da će nekako, ako koristite računala,

it's all mindless button-pushing,

to sve biti bezumno stiskanje gumba,

but if you do it by hand,

ali ako to napravite ručno,

it's all intellectual.

to će biti inteligentno.

This one kind of annoys me, I must say.

Ovo me malo živcira, moram priznati.

Do we really believe

Da li stvarno vjerujemo

that the math that most people are doing in school

da je matematika koju većina ljudi radi u školama

practically today

danas u praksi

is more than applying procedures

stvarno više od puke primjene procedura

to problems they don't really understand, for reasons they don't get?

u problemima koje oni baš ne razume, zbog razloga koje ne shvaćaju?

I don't think so.

Ja ne mislim tako.

And what's worse, what they're learning there isn't even practically useful anymore.

I ono što je najgore, ono što uče tamo nije više uopće primjenjivo u praksi.

Might have been 50 years ago, but it isn't anymore.

Možda je bilo prije 50 godina, ali više nije.

When they're out of education, they do it on a computer.

Kad završe s obrazovanjem, rade to na računalima.

Just to be clear, I think computers can really help with this problem,

Samo da bude jasno, ja mislim da nam računala mogu stvarno pomoći s ovim problemom,

actually make it more conceptual.

ustvari učiniti ga mnogo slikovitijim.

Now, of course, like any great tool,

Naravno, kao i svaki veliki alat

they can be used completely mindlessly,

može ih se koristi potpuno bez razmišljanja,

like turning everything into a multimedia show,

kao pretvaranje svega u multimedijsku prezentaciju,

like the example I was shown of solving an equation by hand,

kao u primjeru u kojem sam pokazao rješavanje jednadžbe na ruke,

where the computer was the teacher --

gdje je računalo bilo učitelj --

show the student how to manipulate and solve it by hand.

koji pokazuje učeniku kako raditi i rješavati ručno.

This is just nuts.

To je jednostavno glupo.

Why are we using computers to show a student how to solve a problem by hand

Zašto koristiti računala da bi pokazali učeniku kako riješiti problem ručno

that the computer should be doing anyway?

kad bi računalo to svakako trebalo napraviti?

All backwards.

Sve je naopako.

Let me show you

Dopustite mi da vam pokažem

that you can also make problems harder to calculate.

kako možete napraviti problem teži za računanje.

See, normally in school,

Vidite, inače u školi,

you do things like solve quadratic equations.

radite stvari kao što je rješavanje kvadratne jednadžbe.

But you see, when you're using a computer,

Ali kad koristite računala,

you can just substitute.

možete samo zamijeniti.

You can make it a quartic equation. Make it kind of harder, calculating-wise.

Zadajte kvadratnu jednadžu, učinite to težim, pametnim računanjem.

Same principles applied --

Ista se načela koriste --

calculations, harder.

računanje, teže.

And problems in the real world

I problemi u stvarnom svijetu

look nutty and horrible like this.

izgledaju ludo i strašno kao ovo.

They've got hair all over them.

Imaju kvake posvuda.

They're not just simple, dumbed-down things that we see in school math.

Nisu samo jednostavni, glupavi problemi kakve viđamo u školskoj matematici.

And think of the outside world.

I razmišljajte o vanjskom svijetu.

Do we really believe that engineering and biology

Da li stvarno mislimo da su inžinjerstvo i biologija

and all of these other things

i sve druge stvari

that have so benefited from computers and maths

koje su toliko dobile zbog računala i matematike

have somehow conceptually gotten reduced by using computers?

nekako izgubile na kvaliteti zbog korištenja računala?

I don't think so -- quite the opposite.

Ja ne mislim tako; baš suprotno.

So the problem we've really got in math education

Dakle problem koji stvarno imamo u matematičkom obrazovanju

is not that computers might dumb it down,

nije u tome da računala mogu zaglupiti,

but that we have dumbed-down problems right now.

nego u tome da smo sada zaglupili probleme.

Well, another issue people bring up

Također, još jedan problem kojeg ljudi iznose

is somehow that hand calculating procedures

je da nekako postupak ručnog računanja

teach understanding.

uči razumijevanju.

So if you go through lots of examples,

Dakle, ako si prošao dosta primjera,

you can get the answer,

možeš doći do odgovora --

you can understand how the basics of the system work better.

možeš bolje razumijeti kako rade osnove sustava.

I think there is one thing that I think very valid here,

Mislim kako je samo jedna stvar tu valjana,

which is that I think understanding procedures and processes is important.

a to je da ja mislim kako su postupci i procesi razumijevanja veoma važni.

But there's a fantastic way to do that in the modern world.

Ali postoji sjajan način kako to postići u današnjem svijetu.

It's called programming.

Zove se programiranje.

Programming is how most procedures and processes

Programiranje je način na koji se većina postupaka i procesa

get written down these days,

danas zapisuje,

and it's also a great way

i to je također sjajan način

to engage students much more

da se uključi studente još više

and to check they really understand.

i da se provjeri da li stvarno razumiju.

If you really want to check you understand math

Ako stvarno želiš provjeriti da li razumiješ matematiku

then write a program to do it.

tada napravi program koji to radi.

So programming is the way I think we should be doing that.

Dakle, programiranje je način na koji bi trebali podučavati.

So to be clear, what I really am suggesting here

Dakle, da budemo jasni, ono što ja stvarno ovdje predlažem

is we have a unique opportunity

je da mi imamo jedinstvenu priliku

to make maths both more practical

učiniti matematiku praktičnijom

and more conceptual, simultaneously.

i primjenjivijom, u isti tren.

I can't think of any other subject where that's recently been possible.

Ne pada mi na pamet ni jedan drugi predmet gdje je to bilo moguće u posljednje vrijeme.

It's usually some kind of choice

To je obično neki odabir

between the vocational and the intellectual.

između stručnog i inteligentnog.

But I think we can do both at the same time here.

Ali ja mislim da možemo imati oboje istovremeno.

And we open up so many more possibilities.

A možemo otvoriti i još mnogo mogućnosti.

You can do so many more problems.

Može se riješiti mnogo više problema.

What I really think we gain from this

Ono što ja stvarno mislim da mi dobivamo od ovog

is students getting intuition and experience

su studenti koji razvijaju intuiciju i iskustvo

in far greater quantities than they've ever got before.

u puno većoj količini nego što su ikad prije imali.

And experience of harder problems --

I iskustvo težih problema --

being able to play with the math, interact with it,

mogućnost igranja s matematikom, korištenja,

feel it.

osjetiti ju.

We want people who can feel the math instinctively.

Mi želimo ljude koji osjećaju matematiku instiktivno.

That's what computers allow us to do.

To je ono što nam računala omogućavaju.

Another thing it allows us to do is reorder the curriculum.

Dodatna stvar koju nam omogućava je preraspodijela programa.

Traditionally it's been by how difficult it is to calculate,

Tradicionalno je to bilo moguće u ovisnosti koliko je bilo teško računanje,

but now we can reorder it

ali sad to možemo promijeniti

by how difficult it is to understand the concepts,

u ovisnosti koliko je teško razumjeti koncept,

however hard the calculating.

i ovisno koliko je teško izračunati.

So calculus has traditionally been taught very late.

Matematička algebra se tradicionalno predaje dosta kasno.

Why is this?

Zašto je to tako?

Well, it's damn hard doing the calculations, that's the problem.

Problem je u težini ručnog računanja.

But actually many of the concepts

Ali zapravo mnogi koncepti

are amenable to a much younger age group.

se mogu primjeniti mnogo ranije.

This was an example I built for my daughter.

To je primjer koji sam ja napravio za svoju kćer.

And very, very simple.

I jako, jako jednostavan.

We were talking about what happens

Razgovarali smo o tome što se dogodi

when you increase the number of sides of a polygon

kad povećaš broj stranica poligona

to a very large number.

do jako velikog broja.

And of course, it turns into a circle.

Naravno, pretvori se u krug.

And by the way, she was also very insistent

Između ostalog, ona ja bila veoma inzistirala

on being able to change the color,

na tome da mijenja i boju,

an important feature for this demonstration.

jako zanimljiva značajka ove demonstracije.

You can see that this is a very early step

Možete vidjeti kako je ovo jako rani korak

into limits and differential calculus

u granice i diferencijalni račun

and what happens when you take things to an extreme --

i što se dogodi kad dovedeš stvari do ekstrema --

and very small sides and a very large number of sides.

jako mali broj stranica i jako velik broj stranica.

Very simple example.

Jako jednostavan primjer.

That's a view of the world

To je pogled na svijet

that we don't usually give people for many, many years after this.

kojeg mi obično ne dajemo ljudima još dugo, dugo godina nakon ovog uzrasta.

And yet, that's a really important practical view of the world.

Konačno, to je jako bitan praktičan pogled na svijet.

So one of the roadblocks we have

Dakle jedna od prepreka koju imamo

in moving this agenda forward

u provođenju ovog plana

is exams.

su ispiti.

In the end, if we test everyone by hand in exams,

Na kraju krajeva, ako testiramo sve ručno u ispitu,

it's kind of hard to get the curricula changed

teško je postići promjenu programa rada

to a point where they can use computers

do točke u kojoj svi mogu koristiti računala

during the semesters.

tokom semestra.

And one of the reasons it's so important --

A to je jedan od razloga zbog kojih je jako važno --

so it's very important to get computers in exams.

jako je važno uvesti računala na ispite.

And then we can ask questions, real questions,

Tada mi možemo postavljati pitanja, prava pitanja,

questions like, what's the best life insurance policy to get? --

pitanja kao što je, koja je osiguravajuća polica najbolja? --

real questions that people have in their everyday lives.

stvarna pitanja s kojima se ljudi svakodnevno susreću.

And you see, this isn't some dumbed-down model here.

Vidite, to nije neki glupi primjer.

This is an actual model where we can be asked to optimize what happens.

To je jedan stvaran primjer gdje nas se može pitati da optimiziramo što se događa.

How many years of protection do I need?

Koliko godina zaštite nam treba?

What does that do to the payments

Što to čini s plaćanjem

and to the interest rates and so forth?

i kamatnim stopama i tako dalje?

Now I'm not for one minute suggesting it's the only kind of question

Sad ja ne navodim uopće kako je ovo jedina vrsta pitanja

that should be asked in exams,

koju bi se trebalo pitati na ispitima,

but I think it's a very important type

ali mislim da je jako važan tip

that right now just gets completely ignored

koji se sada potpuno igonorira

and is critical for people's real understanding.

i ključan je za ljudsko razumijevanje.

So I believe [there is] critical reform

Mislim kako je kritična promjena

we have to do in computer-based math.

koju moramo učiniti u računalno-baziranoj matematici.

We have got to make sure

Moramo osigurati

that we can move our economies forward,

da možemo pokrenuti naprijed našu ekonomiju,

and also our societies,

i naše društvo,

based on the idea that people can really feel mathematics.

oslanjajući se na ideju da ljudi mogu stvarno osjetiti matematiku.

This isn't some optional extra.

To nije neki dodatan izbor.

And the country that does this first

I zemlja koja to učini prva

will, in my view, leapfrog others

će, to je moj pogled, preskočiti druge

in achieving a new economy even,

u postizanju nove ravnomjerne ekonomije,

an improved economy,

unaprijeđene ekonomije,

an improved outlook.

jednog naprednog stajališta.

In fact, I even talk about us moving

U stvari, ja čak govorim o nama koji se krećemo

from what we often call now the "knowledge economy"

od onog što često nazivamo gospodarstvo temeljeno na znanju

to what we might call a "computational knowledge economy,"

u ono što bi mogli nazvati gospodarstvo temeljeno na računalnom znanju,

where high-level math is integral to what everyone does

gdje je visoka razina matematike dio svačijeg posla

in the way that knowledge currently is.

na način na koji je znanje trenutačno ugrađeno.

We can engage so many more students with this,

Možemo uključiti mnogo više studenata s ovim,

and they can have a better time doing it.

i mogu imati ugodnije vrijeme s tim.

And let's understand:

Ajmo pokušati shvatiti,

this is not an incremental sort of change.

ovo nije mala vrsta promjene.

We're trying to cross the chasm here

Pokušavamo premostiti jaz

between school math and the real-world math.

između matematike u školama i matematike u stvarnom svijetu.

And you know if you walk across a chasm,

I znate da ako se šetate preko jaza,

you end up making it worse than if you didn't start at all --

završite tako da ste napravili još gore nego da niste ni počinjali --

bigger disaster.

veću štetu.

No, what I'm suggesting

Ne, ono što ja predlažem

is that we should leap off,

je da mi preskočimo,

we should increase our velocity

povećamo našu brzinu

so it's high,

jako visoko,

and we should leap off one side and go the other --

i preskočimo s jedne strane na drugu --

of course, having calculated our differential equation very carefully.

naravno, trebamo izračunati našu diferencijalnu jednadžbu jako pažljivo.

(Laughter)

(Smijeh)

So I want to see

Dakle, ja želim vidjeti

a completely renewed, changed math curriculum

potpuno nov, promijenjen matematički program

built from the ground up,

izrađen od temelja,

based on computers being there,

temeljen na računalima,

computers that are now ubiquitous almost.

računalima koja su sad sveprisutna.

Calculating machines are everywhere

Računalni strojevi su sad posvuda

and will be completely everywhere in a small number of years.

i bit će sad u potpunosti svugdje u kratkom vremenskom periodu.

Now I'm not even sure if we should brand the subject as math,

Sad više nisam siguran trebamo li nazvati predmet uopće matematika,

but what I am sure is

ali ono što sam siguran je

it's the mainstream subject of the future.

da je ovo temeljni predmet budućnosti.

Let's go for it,

Krenimo do njega.

and while we're about it,

I dok mu se približavamo,

let's have a bit of fun,

zabavimo se,

for us, for the students and for TED here.

za nas, za studente i za TED ovdje.

Thanks.

Hvala.

(Applause)

(Pljesak)