Conrad Wolfram: Teaching kids real math with computers

Prevodilac: Elizabeta Petrovic Lektor: Ivana Korom

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

Trenutno imamo realni problem sa matematičkim obrazovanjem.

Basically, no one's very happy.

U osnovi, niko nije srećan.

Those learning it

Oni koji je uče

think it's disconnected,

misle da je nepovezana,

uninteresting and hard.

nezanimljiva i teška.

Those trying to employ them

Oni koji pokušavaju da ih zaposle

think they don't know enough.

misle da ne znaju dovoljno.

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

Vlade shvataju da je to velika stvar za naše ekonomje,

but don't know how to fix it.

ali ne znaju kako da je poprave.

And teachers are also frustrated.

I nastavnici su takođe frustrirani.

Yet math is more important to the world

Ipak matematika je bitnija svetu

than at any point in human history.

nego u bilo kom momentu u ljudskoj istoriji.

So at one end we've got falling interest

Tako da s jedne strane imamo pad interesovanja

in education in math,

u obrazovanju za matematiku,

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

a sa druge strane imamo više matematički svet,

a more quantitative world than we ever have had.

više kvantitativni svet, nego ikada.

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

Pa koji je problem, zašto se stvorio ovaj jaz,

and what can we do to fix it?

i kako možemo da ga popravimo?

Well actually, I think the answer

Pa ustvari, mislim da nas odgovor

is staring us right in the face:

gleda pravo u lice.

Use computers.

Koristite kompjutere.

I believe

Verujem

that correctly using computers

da je pravilno korišćenje kompjutera

is the silver bullet

srebrni metak

for making math education work.

za uspeh matematičkog obrazovanja.

So to explain that,

Da objasnim to,

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

prvo da vam malo kažem kako matematika izgleda u realnom svetu

and what it looks like in education.

i kako izgleda u obrazovanju.

See, in the real world

Vidite, u realnom svetu

math isn't necessarily done by mathematicians.

matematiku ne rade samo matematičari.

It's done by geologists,

Koriste je geolozi,

engineers, biologists,

inženjeri, biolozi,

all sorts of different people --

svakavi ljudi --

modeling and simulation.

modeliraju i sumuliraju.

It's actually very popular.

Ustvari je veoma popularna.

But in education it looks very different --

Ali u obrazovanju izgleda mnogo drugačije,

dumbed-down problems, lots of calculating,

glupavi problemi, mnogo računanja --

mostly by hand.

uglavnom ručno.

Lots of things that seem simple

Mnoge stvari koje izgledaju jednostavno

and not difficult like in the real world,

i nisu teške kao u realnom svetu,

except if you're learning it.

osim ako ih učite.

And another thing about math:

I još jedna stvar oko matematike:

math sometimes looks like math --

matematika nekad izgleda kao matematika --

like in this example here --

kao u ovom primeru ovde --

and sometimes it doesn't --

a nekad ne izgleda --

like "Am I drunk?"

kao "Da li sam pijan?"

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

I dobijete odgovor koji je kvantitativan u modernom svetu.

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

Ne biste to očekivali pre nekoliko godina.

But now you can find out all about --

Ali sada možete sve da saznate o --

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

na nesreću, moja težina je malo veća od toga, ali --

all about what happens.

svemu šta se događa.

So let's zoom out a bit and ask,

Udaljimo se malo i pitajmo,

why are we teaching people math?

zašto predajemo matematiku?

What's the point of teaching people math?

Koje je poenta učiti ljude matematiku?

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

I posebno, zašto ih uopšte učimo matemaiku?

Why is it such an important part of education

Zašto je toliko važan deo u obrazovanju

as a sort of compulsory subject?

kao obavezan predmet?

Well, I think there are about three reasons:

Mislim da ima 3 razloga:

technical jobs

tehnički poslovi

so critical to the development of our economies,

od velike važnosti za razvoj naših ekonomija,

what I call "everyday living" --

što zovem svakodnevni život.

to function in the world today,

Da biste funkcionisali svakog dana,

you've got to be pretty quantitative,

morate da budete kvantitativni,

much more so than a few years ago:

mnogo više nego pre nekoliko godina.

figure out your mortgages,

Da shvatimo naše hipoteke,

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

da budemo skeptični o vladinim statistikama i takve stvari.

and thirdly, what I would call something like

I treće, što bih nazvao

logical mind training, logical thinking.

treniranje logike uma, logičko razmišljanje.

Over the years

Tokom godina

we've put so much in society

dosta smo uložili u društvo

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

da bude sposobno da radi i misli logički; to je deo ljudskog društva.

It's very important to learn that

Veoma je bitno naučiti to.

math is a great way to do that.

Matematika je super način za to.

So let's ask another question.

Da pitamo drugo pitanje.

What is math?

Šta je matematika?

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

Šta mislimo kada kažemo da radimo matematiku,

or educating people to do math?

ili edukujemo ljude da rade matematiku?

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

Pa mislim da ima 4 koraka, grubo rečeno,

starting with posing the right question.

počevši sa pravim pitanjem.

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

Šta hoćemo da pitamo? Šta pokušavamo da saznamo?

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

I ova stvar je najzeznutija u spoljašnjem svetu,

beyond virtually any other part of doing math.

više od skoro bilo kog dela rađenja matematike.

People ask the wrong question,

Ljudi postavljaju pogrešno pitanje,

and surprisingly enough, they get the wrong answer,

i dovoljno iznenađujuće, dobijaju pogrešan odgovor,

for that reason, if not for others.

bar zbog toga, ako ni zbog čega drugog.

So the next thing is take that problem

Tako da je sledeća stvar da taj problem

and turn it from a real world problem

prebacimo od problema stvarnog sveta

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.

I kada uradite to, tu je onda računanje.

Turn it from that into some answer

Napravite to u neki odgovor

in a mathematical form.

u matematičkoj formi.

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

I naravno, matematika je tu moćna.

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

I onda konačno, prebacite je u realni svet.

Did it answer the question?

Da li je to odgovorilo na pitanje?

And also verify it -- crucial step.

I proverite -- krucijalni korak.

Now here's the crazy thing right now.

Sada ide luda stvar.

In math education,

U matematičkom obrazovanju,

we're spending about perhaps 80 percent of the time

trošimo oko 80% vremena

teaching people to do step three by hand.

učeći ljude da ručno rade 3. korak.

Yet, that's the one step computers can do

Ipak, to je jedan korak koji kompjuteri mogu da rade

better than any human after years of practice.

bolje nego bilo koji čovek posle godina prakse.

Instead, we ought to be using computers

Umesto toga, moramo da koristimo kompjutere

to do step three

za ovaj treći korak

and using the students to spend much more effort

i nateramo učenike da se više trude

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

oko učenja prvog koraka, drugog i četvrtog --

conceptualizing problems, applying them,

konceptualizovanja problema, njihove primene,

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

naterati nastavnika da ih nauči tome.

See, crucial point here:

Vidite, bitna stavka ovde:

math is not equal to calculating.

matematika i računanje nije isto.

Math is a much broader subject than calculating.

Matematika je širi predmet od računanja.

Now it's understandable that this has all got intertwined

Sada je razumljivo da se ovo izmešalo

over hundreds of years.

tokom stotina godina.

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

I postojao je samo jedan način računanja i to ručno.

But in the last few decades

Ali poslednjih nekoliko decenija

that has totally changed.

ovo se skroz promenilo.

We've had the biggest transformation of any ancient subject

Imamo najveću transformaciju od svih drevnih predmeta

that I could ever imagine with computers.

koju bih ikada mogao da zamislim sa kompjuterom.

Calculating was typically the limiting step,

Računanje je bio tipični ograničavajući korak,

and now often it isn't.

a on retko to jeste.

So I think in terms of the fact that math

Pa po pitanju matematike

has been liberated from calculating.

ona je oslobođena računanja.

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

Ali njeno oslobađanje još nije ušlo u obrazovanje.

See, I think of calculating, in a sense,

Vidite, mislim o računanju kao

as the machinery of math.

o mašini matematike.

It's the chore.

To je zadatak.

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

I stvar koju biste izbegavali, ostavili biste mašini da radi.

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

To je sredstvo za postizanje cilja, a ne cilj.

and automation allows us

I automatizacija nam dozvoljava

to have that machinery.

da imamo tu mašinu.

Computers allow us to do that --

Kompjuteri nam dozvoljavaju to.

and this is not a small problem by any means.

I ovo uopšte nije mali problem.

I estimated that, just today, across the world,

I procenjujem da, samo danas širom sveta,

we spent about 106 average world lifetimes

trošimo u proseku oko 106 svetskih života

teaching people how to calculate by hand.

u učenju ljudi kako da računaju ručno.

That's an amazing amount of human endeavor.

To je mnogo ljudskog napora.

So we better be damn sure --

I bolje bi nam bilo da budemo sigurni --

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

i usput, većini njih i nije baš bilo zabavno.

so we better be damn sure

I bolje bi bilo da budemo sigurni

that we know why we're doing that

da znamo zašto to radimo

and it has a real purpose.

i da to ima pravu svrhu.

I think we should be assuming computers

I mislim da bi trebalo da koristimo kompjutere

for doing the calculating

za računanje

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

i računamo ručno samo kada ima smisla tome učiti ljude.

And I think there are some cases.

I mislim da ima nekoliko slučajeva.

For example: mental arithmetic.

Na primer: mentalna aritmetika.

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

I dalje radim puno toga, uglavnom za procenu.

People say, "Is such and such true?"

Ljudi kažu, da li je to i to tačno,

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

i ja kažem, hmm, nisam siguran. Grubo ću razmisliti o tome.

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

Brže je i praktičnije.

So I think practicality is one case

I mislim da je praktičnost jedan slučaj

where it's worth teaching people by hand.

gde vredi učiti ljude ručnom računanju.

And then there are certain conceptual things

Ali onda postoje određene konceptualne stvari

that can also benefit from hand calculating,

koje imaju korist od ručnog računanja,

but I think they're relatively small in number.

ali mislim da ih malo ima.

One thing I often ask about

I često pitam o

is ancient Greek and how this relates.

staro-grčkom i kako je to povezano.

See, the thing we're doing right now

Vidite, stvar koju sad radimo,

is we're forcing people to learn mathematics.

je da teramo ljude 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

I uopšte ne predlažem da ako ljudi vole ručno računanje

or in following their own interests

ili prate svoje interese

in any subject however bizarre --

u bilo kom predmetu, nebitno koliko bizarno --

they should do that.

da treba to da rade.

That's absolutely the right thing,

To je apsolutno prava stvar,

for people to follow their self-interest.

da ljudi prate svoje interese.

I was somewhat interested in ancient Greek,

I bio sam zainteresovan za staro-grčki,

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

ali mislim da ne treba da teramo celu populaciju

to learn a subject like ancient Greek.

da uče predmete kao staro-grčki.

I don't think it's warranted.

Mislim da nema garanciju.

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

Tako da imamo razliku između toga šta teramo ljude da rade

and the subject that's sort of mainstream

i predmeta koji je jedan od bitnijih

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

i predmeta koji bi ljudi mogli da prate zbog svog interesovanja

and perhaps even be spiked into doing that.

i možda budu naterani da to rade.

So what are the issues people bring up with this?

I koja pitanja imaju ljudi s ovim?

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

Pa jedno od njih je, oni kažu, treba prvo naučiti osnovu.

You shouldn't use the machine

Ne treba da se koriste mašine

until you get the basics of the subject.

dok se ne shvati osnova predmeta.

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

Pa je moje uobičajeno pitanje, šta mislite pod osnovom?

Basics of what?

Osnova 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?

učiti kako da ga popravimo, dizajniramo?

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

Da li je osnova pisanja oštrenje pera?

I don't think so.

Ne mislim tako.

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

Mislim da treba da odvojite osnovu onog što pokušavate da naučite

from how it gets done

od toga kako se radi

and the machinery of how it gets done

i mašine kojom se radi.

and automation allows you to make that separation.

I automatizacija nam dozvoljava to razdvajanje.

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

I pre stotinu godina, sigurno je tačno da da biste vozili kola

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

morali ste da znate dosta o mehanici kola

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

i kako vreme ubrzanja radi i te stvari.

But automation in cars

Ali automatizacija u kolima

allowed that to separate,

nam dozvoljava da to razdvojimo,

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

tako da je vožnja sada totalno odvojeni predmet, takoreći,

from engineering of the car

od inženjeringa auta

or learning how to service it.

ili učenja o njegovom servisu.

So automation allows this separation

Tako da automatizacija omogućava ovo razdvajanje

and also allows -- in the case of driving,

i takođe -- u slučaju vožnje,

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

i verujem da u budućnost matematike --

a democratized way of doing that.

demokratski način da se to uradi.

It can be spread across a much larger number of people

Može da se raširi na veći broj ljudi

who can really work with that.

koji se stvarno bave time.

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

Tako da postoji još jedna stvar sa osnovom.

People confuse, in my view,

Ljude zbunjuje, po meni,

the order of the invention of the tools

red inovacije alata

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

sa redom kojim se oni koriste pri predavanju.

So just because paper was invented before computers,

Tako da samo zato što je papir izumljen pre kompjutera,

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

ne znači da ćete bolje savladati osnovu predmeta

by using paper instead of a computer

na papiru nego na kompjuteru

to teach mathematics.

dok predajete matematiku.

My daughter gave me a rather nice anecdote on this.

Moja ćerka mi je dala finu anegdotu o ovome.

She enjoys making what she calls "paper laptops."

Ona uživa praveći takozvane papirne laptopove.

(Laughter)

(Smeh)

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

I pitao sam je jednog dana: "Znaš, kada sam bio tvojih godina,

I didn't make these.

nisam pravio ovo.

Why do you think that was?"

Zašto misliš da je tako?"

And after a second or two, carefully reflecting,

I razmišljajući pažljivo posle sekunde ili dve,

she said, "No paper?"

rekla je: "Niste imali papir?"

(Laughter)

(Smeh)

If you were born after computers and paper,

Ako ste rođeni nakon kompjutera i papira,

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

nije bitno kojim ste redom učeni njima,

you just want to have the best tool.

vi samo želite najbolji alat.

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

I još jedna stvar: "kompjuteri zatupljuju matematku."

That somehow, if you use a computer,

Da je nekako, ako koristite kompjuter,

it's all mindless button-pushing,

to sve bezumno pritiskanje dugmeta,

but if you do it by hand,

al' ako radite ručno,

it's all intellectual.

onda je intelektualno.

This one kind of annoys me, I must say.

Ovo me malo nervira, moram da kažem.

Do we really believe

Da li stvarno verujemo

that the math that most people are doing in school

da je matematika koju većina ljudi radi u školi

practically today

danas praktično

is more than applying procedures

više nego primenjivanje procedura

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

na probleme koje ne razumeju, zbog razloga koje ne razumeju?

I don't think so.

Ne mislim tako.

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

Još gore je što ono što uče skoro da nije više korisno.

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

Možda je bilo pre 50 godina, ali nije više.

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

Kada završe školovanje, oni to rade na kompjuteru.

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

Samo da razjasnim, mislim da kompjuteri mogu stvarno da pomognu sa ovim problemom,

actually make it more conceptual.

da ga naprave konceptualnijim.

Now, of course, like any great tool,

Naravno, kao i svaki super alat

they can be used completely mindlessly,

oni mogu da se koriste totalno bez razmišljanja,

like turning everything into a multimedia show,

kao pretvaranje svega u multimedijalni šou,

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

kao primer koji sam vam pokazao o ručnom rešavanju jednačine,

where the computer was the teacher --

gde je kompjuter bio nastavnik --

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

pokazao učeniku kako da manipuliše i ručno reši.

This is just nuts.

To je ludo.

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

Zašto koristimo kompjutere da pokažemo učenicima kako da problem

that the computer should be doing anyway?

koji bi ionako kompjuter trebalo da reši?

All backwards.

Sve ide unazad.

Let me show you

Pokazaću vam

that you can also make problems harder to calculate.

da možete da napravite probleme koji se teže računaju.

See, normally in school,

Vidite, normalno u školi,

you do things like solve quadratic equations.

rešavate kvadratne jednačine.

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

Ali kada koristite kompjuter,

you can just substitute.

možete samo da zamenite.

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

Napravite jednačinu na četvrti stepen; malo težu, u smislu računanja.

Same principles applied --

Isti principi se primenjuju --

calculations, harder.

računanje, teže.

And problems in the real world

I problemi u realnom svetu

look nutty and horrible like this.

izgledaju ludo i strašno kao ovo.

They've got hair all over them.

Čupavi su svuda.

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

Nisu samo jednostavne, glupave stvari koje vidimo u školskoj matematici.

And think of the outside world.

I razmislite o spoljnom svetu.

Do we really believe that engineering and biology

Da li stvarno verujemo da su se inženjerstvo

and all of these other things

i biologija i sve te stvari

that have so benefited from computers and maths

koje dobro koriste kompjutere i matematiku

have somehow conceptually gotten reduced by using computers?

nekako konceptualno osiromašile, korišćenjem kompjutera?

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

Ne mislim tako, već suprotno.

So the problem we've really got in math education

Tako da problem koji stvarno imamo u matematičkom obrazovanju

is not that computers might dumb it down,

nije da su je kompjuteri zatupili,

but that we have dumbed-down problems right now.

nego da smo mi sada zatupili probleme.

Well, another issue people bring up

Pa, još jedna stvar koju ljudi spominju

is somehow that hand calculating procedures

je da nekako proces ručnog računanja

teach understanding.

uči razumevanje.

So if you go through lots of examples,

I ako prođete kroz dosta primera,

you can get the answer,

možete da dobijete odgovor --

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

možete da razumete kako osnova sistema bolje radi.

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

Mislim da je jedna stvar ovde validna,

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

a to je da je važno razumevanje procedura i procesa.

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

Ali postoji fantastičan način da uradimo to u modernom svetu.

It's called programming.

To se zove programiranje.

Programming is how most procedures and processes

Programiranje je način kako se većina procedura

get written down these days,

i procesa piše ovih dana,

and it's also a great way

i takođe je super način

to engage students much more

da se više angažuju učenici

and to check they really understand.

da bi proverili da stvarno razumeju.

If you really want to check you understand math

Ako stvarno želite da proverite da razumete matematiku

then write a program to do it.

onda napišite program za to.

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

Tako da je programiranje način na koji to treba raditi.

So to be clear, what I really am suggesting here

Da budemo jasni, ono što predlažem je

is we have a unique opportunity

da imamo jedinstvenu priliku

to make maths both more practical

da napravimo matematiku i praktičnijom

and more conceptual, simultaneously.

i konceptualnijom, istovremeno.

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

Ne mogu da smislim nijedan drugi predmet kod koga je to bilo skoro moguće.

It's usually some kind of choice

To je obično neka vrsta izbora

between the vocational and the intellectual.

između radnog i intelektualnog.

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

Ali mislim da možemo da uradimo oba istovremeno.

And we open up so many more possibilities.

I otvaramo toliko mnogo mogućnosti.

You can do so many more problems.

Možete da rešavate mnogo više problema.

What I really think we gain from this

I ono što mislim da dobijamo iz ovoga

is students getting intuition and experience

je da učenici dobijaju intuiciju i iskustvo

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

u većim količinama nego ikada pre.

And experience of harder problems --

I iskustvo težih problema --

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

biti sposoban igrati se sa matematikom, biti u interakciji

feel it.

s njom, osetiti je.

We want people who can feel the math instinctively.

Želimo ljude koji mogu da instiktivno osete matematiku.

That's what computers allow us to do.

To je ono što nam kompjuteri omogućavaju.

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

Još jedna stvar koju nam omogućava je da reorganizujemo program.

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

Tradicionalno bilo je bitno koliko je teško izračunati,

but now we can reorder it

ali sada možemo da reorganizujemo

by how difficult it is to understand the concepts,

prema tome koliko je teško razumeti koncept,

however hard the calculating.

nebitno koliko je teško izračunati.

So calculus has traditionally been taught very late.

Tako da se računanje tradicionalno uči veoma kasno.

Why is this?

Zašto?

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

Pa, veoma je teško, to je problem.

But actually many of the concepts

Ali ustvari, mnogi od koncepata

are amenable to a much younger age group.

su prikladni za mnogo mlađe grupe.

This was an example I built for my daughter.

Ovo je primer koji sam napravio za moju ćerku.

And very, very simple.

I veoma, veoma jednostavan.

We were talking about what happens

Pričali smo šta se dešava

when you increase the number of sides of a polygon

kada povećate broj strana poligona

to a very large number.

na veoma veliki broj.

And of course, it turns into a circle.

I naravno, on postaje krug.

And by the way, she was also very insistent

I usput, bila je veoma uporna

on being able to change the color,

da promeni boju,

an important feature for this demonstration.

što je najbitnija stvar demonstracije.

You can see that this is a very early step

Vidite da je ovo početni korak

into limits and differential calculus

u graničnim vrednostima i deferencijalnim računima

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

i šta se dešava kada dovedete stvari do ekstrema --

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

i veoma male strane i veliki broj strana.

Very simple example.

To je veoma jednostavan primer.

That's a view of the world

To je viđenje sveta

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

koje obično ne dajemo ljudima mnogo, mnogo godina posle ovoga.

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

A ipak, to je veoma bitno praktično viđenje sveta.

So one of the roadblocks we have

Jedna od barijera koje imamo

in moving this agenda forward

na putu razvoja ovog plana

is exams.

su testovi.

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

Na kraju, ako testiramo svakog ručno na ispitu,

it's kind of hard to get the curricula changed

onda je teško promeniti program

to a point where they can use computers

u tačku gde mogu da koriste kompjutere

during the semesters.

tokom polugodišta.

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

I jedan od razloga zašto je važno --

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

veoma je bitno imati kompjutere na ispitima.

And then we can ask questions, real questions,

I onda možemo da postavljamo pitanja, realna pitanja,

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

kao, koja je najbolja polisa osiguranja? --

real questions that people have in their everyday lives.

realna pitanja koja ljudi imaju u svakodnevnom životu.

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

I vidite, ovo nije zatupljujući model.

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

Ovo je pravi model gde možemo da optimizujemo šta se događa.

How many years of protection do I need?

Koliko godina zaštite mi treba?

What does that do to the payments

Šta to čini isplatama

and to the interest rates and so forth?

i kamatama i tako dalje?

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

I ne predlažem da je ovo jedina vrsta pitanja

that should be asked in exams,

koje bi trebalo postaviti na testovima,

but I think it's a very important type

ali mislim da je bitan tip

that right now just gets completely ignored

koji je sada kompletno ignorisan

and is critical for people's real understanding.

i kritičan za ljudsko stvarno razumevanje.

So I believe [there is] critical reform

Tako da verujem da kritičnu reformu

we have to do in computer-based math.

treba da radimo matematici zasnovanoj na kompjuteru.

We have got to make sure

Moramo da budemo sigurni

that we can move our economies forward,

da možemo da unapredimo ekonomije,

and also our societies,

i naša društva,

based on the idea that people can really feel mathematics.

na ideji da ljudi stvarno mogu da osete matematiku.

This isn't some optional extra.

To nije neki opcioni višak.

And the country that does this first

I zemlja koja prva ovo uradi

will, in my view, leapfrog others

će, po meni, preskočiti druge

in achieving a new economy even,

u postizanju čak i nove ekonomije,

an improved economy,

poboljšane ekonomije,

an improved outlook.

poboljšanih ishoda.

In fact, I even talk about us moving

Ustvari, pričam o tome da se pomerimo

from what we often call now the "knowledge economy"

od toga što često zovemo ekonomija znanja

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

do toga što bismo zvali kompjuterizovanja ekonomija znanja,

where high-level math is integral to what everyone does

gde je viša matematika sastavni deo svega što ljudi rade

in the way that knowledge currently is.

na način na koji znanje trenutno funkcioniše.

We can engage so many more students with this,

Možemo da uključimo mnogo više učenika s ovim,

and they can have a better time doing it.

i mogu da provode bolje vreme radeći to.

And let's understand:

I razumimo,

this is not an incremental sort of change.

ovo nije rastuća promena.

We're trying to cross the chasm here

Pokušavamo da preskočimo jaz ovde

between school math and the real-world math.

između školske matematike i matematike u realnom svetu.

And you know if you walk across a chasm,

I znate ako hodate preko jaza, onda

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

je samo još gore nego da niste ni krenuli -

bigger disaster.

veća katastrofa.

No, what I'm suggesting

Ali, ja predlažem

is that we should leap off,

da bi trebalo da preskočimo,

we should increase our velocity

da povećamo našu brzinu,

so it's high,

da bude visoka,

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

i treba da preskočimo s jedne strane na drugu --

of course, having calculated our differential equation very carefully.

naravno, ako smo pažljivo računali naše diferencijalne jednačine.

(Laughter)

(Smeh)

So I want to see

I želim da vidim

a completely renewed, changed math curriculum

kompletno novi, promenjeni matematički program,

built from the ground up,

sagrađen iz temelja,

based on computers being there,

zasnovan na kompjuterima,

computers that are now ubiquitous almost.

kompjuterima koji su sada skoro svuda prisutni.

Calculating machines are everywhere

Računari su svuda

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

i biće potpuno svuda za nekoliko godina.

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

Nisam siguran da bi trebalo da brendiramo predmet kao matematika,

but what I am sure is

ali ono u šta sam siguran je

it's the mainstream subject of the future.

da je ovo glavni predmet budućnosti.

Let's go for it,

Uradimo to.

and while we're about it,

I dok smo tu,

let's have a bit of fun,

možemo da se malo zabavimo,

for us, for the students and for TED here.

za nas, za učenike i za TED.

Thanks.

Hvala.

(Applause)

(Aplauz)