Math class needs a makeover | Dan Meyer

Prevoditelj: Mislav Ante Omazić - EFZG Recezent: Predrag Pale

Can I ask you to please recall a time

Mogu li vas zamoliti da se prisjetite vremena

when you really loved something --

kada ste uistinu nešto voljeli,

a movie, an album, a song or a book --

film, album, pjesmu ili knjigu,

and you recommended it wholeheartedly

i preporučili ste to iskreno

to someone you also really liked,

nekome do koga vam je stalo,

and you anticipated that reaction, you waited for it,

i očekivali ste tu reakciju, čekali ste na nju,

and it came back, and the person hated it?

i ona je došla; osoba je mrzila vašu preporuku.

So, by way of introduction,

Čisto uvoda radi

that is the exact same state

ovo je upravo stanje

in which I spent every working day of the last six years. (Laughter)

u kojem provodim gotovo svaki dan u posljednjih šest godina.

I teach high school math.

Naime, predajem matematiku u srednjoj školi.

I sell a product to a market

Prodajem proizvod tržištu

that doesn't want it, but is forced by law to buy it.

koje ga ne želi, ali ga je zakonom prisiljeno kupiti.

I mean, it's just a losing proposition.

To je unaprijed izgubljen slučaj.

So there's a useful stereotype about students that I see,

Postoji koristan stereotip o učenicima koji ja vidim,

a useful stereotype about you all.

koristan stereotip o svima vama.

I could give you guys

Mogao bih vam dati

an algebra-two final exam,

završni ispit iz algebre dva

and I would expect no higher

i ne bih očekivao

than a 25 percent pass rate.

veću prolaznost od 25%.

And both of these facts say less about you or my students

Obje ove činjenice govore manje o vama ili mojim učenicima

than they do about what we call math education

nego o onome što nazivamo matematičko obrazovanje

in the U.S. today.

u današnjoj Americi.

To start with, I'd like to break math down into two categories.

Krenuo bih s time da podijelim matematiku na dvije kategorije.

One is computation; this is the stuff you've forgotten.

Jedna je računanje. To su stvari koje ste zaboravili.

For example, factoring quadratics with

Na primjer, rastavljanje kvadrata s

leading coefficients greater than one.

vodećim koeficijentima koji su veći od jedan.

This stuff is also really easy to relearn,

Te je stvari također jako jednostavno ponovno naučiti,

provided you have a really strong grounding

pod uvjetom da imate stvarno jako razvijeno

in reasoning. Math reasoning --

racionalno razmišljanje, matematičko promišljanje.

we'll call it the application

Nazvati ćemo to primjenom

of math processes to the world around us --

matematičkih procesa na svijet oko nas.

this is hard to teach.

To je teško za naučiti nekoga.

This is what we would love students to retain,

To je ono što bismo voljeli da naši učenici dobiju i zadrže,

even if they don't go into mathematical fields.

čak i ako ne nastave obrazovanje u matematičkom području.

This is also something that, the way we teach it in the U.S.

To je također nešto, s obzirom na način na koji poučavamo u Americi

all but ensures they won't retain it.

što oni vrlo vjerojatno neće zadržati.

So, I'd like to talk about why that is,

Tako da ću ja govoriti o tome zašto je to tako,

why that's such a calamity for society, what we can do about it

zašto je to nesreća za društvo, što možemo po tom pitanju učiniti,

and, to close with, why this is an amazing time

i, za kraj, zašto je ovo tako nevjerojatno vrijeme

to be a math teacher.

biti nastavnikom matematike.

So first, five symptoms

Prije svega, pet simptoma

that you're doing math reasoning wrong

koji pokazuju da krivo poučavate matematiku

in your classroom.

u svom razredu.

One is a lack of initiative; your students don't self-start.

Prvi je manjak inicijative; vaši studenti ne započinju sami.

You finish your lecture block

Nakon što završite sa svojim blokom predavanja

and immediately you have five hands going up

imate po pravilu barem pet ruku u zraku

asking you to re-explain the entire thing at their desks.

koje od vas traže da ponovno objasnite cijelu stvar.

Students lack perseverance.

Učenicima nedostaje upornosti.

They lack retention; you find yourself

Njima nedostaje sposobnosti da zapamte; često smo

re-explaining concepts three months later, wholesale.

u situaciji da moram ponovno objašnjavati koncepte tri mjeseca kasnije.

There's an aversion to word problems,

Ne vole problemske zadatke, one s riječima

which describes 99 percent of my students.

a takvih je 99% mojih učenika.

And then the other one percent

A onih 1%

is eagerly looking for the formula

nestrpljivo traži formulu

to apply in that situation.

koja se može primijeniti na zadatak.

This is really destructive.

To je stvarno destruktivno.

David Milch, creator of "Deadwood" and other amazing TV shows,

David Milch, autor serije "Deadwood" i drugih čudesnih TV emisija,

has a really good description for this.

ima stvarno dobar opis ovoga.

He swore off creating

Zarekao se da će stvoriti

contemporary drama,

suvremenu dramu,

shows set in the present day,

koja će se prikazivati u sadašnjosti,

because he saw that when people fill their mind

jer je shvatio da kada ljudi ispune um

with four hours a day of, for example, "Two and a Half Men," no disrespect,

s četiri sada dnevno, na primjer serije "Dva i pol muškarca", uz dužno poštovanje,

it shapes the neural pathways, he said,

oblikuju im se veze u mozgu, kaže,

in such a way that they expect simple problems.

na takav način da očekuju samo jednostavne probleme.

He called it, "an impatience with irresolution."

On je to nazvao, "nestrpljivost za rješenje".

You're impatient with things that don't resolve quickly.

Nestrpljivi ste sa stvarima koje se ne razriješe brzo.

You expect sitcom-sized problems that wrap up in 22 minutes,

Očekujete jednostavne probleme kao u serijama koje stanu u 22 minute,

three commercial breaks and a laugh track.

uključujuće tri pauze za reklame i vrijeme za smijeh.

And I'll put it to all of you,

I neću vam otkriti novost ako vam kažem

what you already know, that no problem worth solving is that simple.

da niti jedan problem vrijedan rješavanja nije jednostavan.

I am very concerned about this

Jako sam zabrinut zbog svega toga,

because I'm going to retire in a world that my students will run.

jer ja ću otići u mirovinu u svijetu koji će voditi moji učenici.

I'm doing bad things

Činim loše stvari

to my own future and well-being

vlastitoj budućnosti i blagostanju

when I teach this way.

kada poučavam na taj način.

I'm here to tell you that the way our textbooks -- particularly

Ovdje sam da vam kažem kako je način na koji naši udžbenici, posebno

mass-adopted textbooks -- teach math reasoning

općeprihvaćeni udžbenici, podučavaju matematičko promišljanje

and patient problem solving,

i strpljivo rješavanje problema,

it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.

funkcionalno jednak kao kada upalite "sapunicu" i mislite da ste riješili problem.

(Laughter)

(Smijeh)

In all seriousness. Here's an example from a physics textbook.

Za ozbiljno, ovdje je primjer iz udžbenika iz fizike.

It applies equally to math.

Jednako je primjenjiv i na matematiku.

Notice, first of all here,

Primjetiti ćete prije svega

that you have exactly three pieces of information there,

kako su dane točno tri informacije ovdje,

each of which will figure into a formula

od kojih će se svaka preobličiti u formulu

somewhere, eventually,

negdje, na kraju,

which the student will then compute.

koju će učenici onda izračunati.

I believe in real life.

Mislim kako u stvarnom životu to ne postoji.

And ask yourself, what problem have you solved, ever,

Zapitajte se: koji ste problem ikada riješili,

that was worth solving

a koji je bio vrijedan rješavanja,

where you knew all of the given information in advance;

gdje ste znali sve ključne informacije unaprijed,

where you didn't have a surplus of information and you had to filter it out,

ili niste imali višak informacija od kojih ste trebali izabrati ključne,

or you didn't have sufficient information

ili niste imali manjak informacija

and had to go find some.

i trebali ste ih pronaći negdje.

I'm sure we all agree that no problem worth solving is like that.

Siguran sam da ćemo se svi složiti kako niti jedan problem vrijedan rješavanja nije takav.

And the textbook, I think, knows how it's hamstringing students

Udžbenik, mislim, zna kako sakati učenike.

because, watch this, this is the practice problem set.

Jer, pogledajte ovo, ovo je praktični zadatak.

When it comes time to do the actual problem set,

Kada dođe vrijeme za rješavanje stvarnih zadataka,

we have problems like this right here

zadaju se zadaci poput ovoga ovdje

where we're just swapping out numbers and tweaking the context a little bit.

gdje samo izmjenimo brojeve i malo promjenimo kontekst.

And if the student still doesn't recognize the stamp this was molded from,

Ako učenik još uvijek ne prepozna po kojoj se šabloni zadatak rješava

it helpfully explains to you

prijateljski vam objasni

what sample problem you can return to to find the formula.

u kojem ranijem primjeru možete pronaći pravu formulu za rješenje.

You could literally, I mean this,

Mogli bi doslovno, to stvarno mislim,

pass this particular unit without knowing any physics,

proći ovo poglavlje bez imalo razumijevanja fizike,

just knowing how to decode a textbook. That's a shame.

samo treba znati kako dešifrirati udžbenik. To je velika šteta.

So I can diagnose the problem a little more specifically in math.

Problem mogu još bolje dijagnosticirati u matematici.

Here's a really cool problem. I like this.

Ovdje je stvarno zgodan zadatak.

It's about defining steepness and slope

Odnosi se na definiranje kosine i nagiba

using a ski lift.

koristeći ski lift.

But what you have here is actually four separate layers,

Ali ono što imate ovdje jesu u stvari četiri odvojene razine.

and I'm curious which of you can see the four separate layers

I baš me zanima tko od vas može vidjeti četiri odvojene razine

and, particularly, how when they're compressed together

i, posebno, kako to što su pomiješane zajedno

and presented to the student all at once,

i predstavljeni učenicima svi odjednom,

how that creates this impatient problem solving.

kako to uzrokuje taj proces nestrpljivog rješavanja problema.

I'll define them here: You have the visual.

Definirati ću ih ovdje. Imate sliku.

You also have the mathematical structure,

Imate i matematičku strukturu

talking about grids, measurements, labels,

govorim o mrežama, mjerama, oznakama,

points, axes, that sort of thing.

točkama, osima, i tako dalje.

You have substeps, which all lead to what we really want to talk about:

Imate međukorake, koji vas vode do onoga o čemu uistinu želimo razgovarati,

which section is the steepest.

koja dio skijaške staze je najstrmiji.

So I hope you can see.

Nadam se da možete vidjeti.

I really hope you can see how what we're doing here

Stvarno se nadam da možete vidjeti, kako

is taking a compelling question, a compelling answer,

uzmijaći izazovno pitanje, izazovni odgovor

but we're paving a smooth, straight path

mi utiremo glatki, ravni put

from one to the other

od jednog prema drugom,

and congratulating our students for how well

i čestitamo našim učenicima koliko dobro

they can step over the small cracks in the way.

mogu savladati male pukotine po tom putu.

That's all we're doing here.

To je ono što radimo.

So I want to put to you that if we can separate these in a different way

Tako vas ja želim potaknuti da razmišljamo na drugačiji način

and build them up with students,

i sagradimo taj put zajedno sa učenicima,

we can have everything we're looking for in terms of patient problem solving.

možemo imati sve ono što želimo s aspekta strpljivog rješavanja zadatka.

So right here I start with the visual,

Ovdje ću započeti sa skicom

and I immediately ask the question:

i odmah vas upitati:

Which section is the steepest?

Koji dio staze je najstrmiji?

And this starts conversation

I time ćemo započeti razogovor

because the visual is created in such a way where you can defend two answers.

jer je skica napravljena na takav način da mogu biti dva odgovora.

So you get people arguing against each other,

Pa će se ljudi prepirati

friend versus friend,

prijatelji protiv prijatelja,

in pairs, journaling, whatever.

u parovima, bilokako.

And then eventually we realize

I onda ćemo na kraju spoznati

it's getting annoying to talk about

postaje zamorno nekome govoriti o

the skier in the lower left-hand side of the screen

skijašu u donjoj lijevoj strani ekrana

or the skier just above the mid line.

ili o skijašu iznad srednje linije.

And we realize how great would it be

I shvatimo koliko bi sjajno bilo

if we just had some A, B, C and D labels

kada bismo imali samo A, B, C i D oznake

to talk about them more easily.

o kojima možemo jednostavno razgovarati.

And then as we start to define what does steepness mean,

A kad počnemo definirati što strmina znači,

we realize it would be nice to have some measurements

shvatimo, kako bi bilo dobro imati neku mjeru

to really narrow it down, specifically what that means.

kako bismo suzili značenje, na specifično značenje.

And then and only then,

I tek tada,

we throw down that mathematical structure.

ubacimo matematičku strukturu.

The math serves the conversation,

Matematika služi razgovoru.

the conversation doesn't serve the math.

Razgovor ne služi matematici.

And at that point, I'll put it to you that nine out of 10 classes

U tom trenutku, uvjeravam vas, 9 od 10 razreda

are good to go on the whole slope, steepness thing.

je sposobno razmatrati problem kosine, strmine.

But if you need to,

Ali ako je potrebno

your students can then develop those substeps together.

vaši učenici mogu onda razviti te podkorake zajedno.

Do you guys see how this, right here, compared to that --

Vidite li vi kako ovo ovdje, uspoređeno s ovime --

which one creates that patient problem solving, that math reasoning?

koji od njih uči, dovodi do strpljivog rješavanja problema, matematičkog promišljanja?

It's been obvious in my practice, to me.

Prema praksi koju imam meni je odgovor očit.

And I'll yield the floor here for a second to Einstein,

I podijeliti ću podij na sekundu s Einsteinom,

who, I believe, has paid his dues.

koji je, vjerujem, to shvatio iskustvom.

He talked about the formulation of a problem being so incredibly important,

On je govorio o tome koliko je iznimno važno formuliranje problema,

and yet in my practice, in the U.S. here,

ipak u mojoj praksi, ovdje u SAD-u,

we just give problems to students;

mi jednostavno dzadajemo učenicima zadatke;

we don't involve them in the formulation of the problem.

ne uključujemo ih u formuliranje problema.

So 90 percent of what I do

Tako je 90% onoga što radim

with my five hours of prep time per week

u svojih pet sati priprema tjedno

is to take fairly compelling elements

jest da uzmem poticajne elemente

of problems like this from my textbook

zadataka poput ovih iz mog udžbenika i preoblikujem ih tako

and rebuild them in a way that supports math reasoning and patient problem solving.

da podržavaju matematičko promišljanje i strpljivo rješavanje problema.

And here's how it works.

I ovdje je prikazano kako to radi.

I like this question. It's about a water tank.

Volim ovakva pitanja. Radi se o spremniku za vodu.

The question is: How long will it take you to fill it up?

Pitanje je: Koliko je vremena potrebno da se napuni? Dobro?

First things first, we eliminate all the substeps.

Kao prvo, eliminiramo sve pod korake.

Students have to develop those,

Učenici ih sami moraju razviti.

they have to formulate those.

Oni ih moraju formulirati.

And then notice that all the information written on there is stuff you'll need.

I sada možete primjetiti kako su sve informacije napisane ovdje one stvari koje trebate.

None of it's a distractor, so we lose that.

Niti jedna od njih ne zbunjuje, tako ćemo ih propustiti.

Students need to decide, "All right, well,

Učenici moraju odlučiti,

does the height matter? Does the side of it matter?

je li visina važna? Je li veličina važna?

Does the color of the valve matter? What matters here?"

Je li važna boja ventila? Što je važno ovdje?

Such an underrepresented question in math curriculum.

Ovo je podcjenjeno pitanje u nastavi matematike.

So now we have a water tank.

Tako imamo spremnik za vodu.

How long will it take you to fill it up? And that's it.

Koliko dugo će trebati da se napuni, i to je to.

And because this is the 21st century

I zato jer je ovo 21. stoljeće,

and we would love to talk about the real world on its own terms,

i zato jer bismo htjeli raspravljati o stvarnom svijetu kakav jest,

not in terms of line art or clip art

a ne kao skica, crtež,

that you so often see in textbooks,

što tako često vidimo u udžbenicima,

we go out and we take a picture of it.

izađemo i fotografiramo ga.

So now we have the real deal.

Sada imamo zadatak iz stvarnog života.

How long will it take it to fill it up?

Koliko je potrebno da se napuni?

And then even better is we take a video,

Ili, čak bolje, napravimo video,

a video of someone filling it up.

snimku nekoga tko ga puni.

And it's filling up slowly, agonizingly slowly.

A puni se polako, agonizirajuće polako.

It's tedious.

To je mučno.

Students are looking at their watches, rolling their eyes,

Učenici gledaju na svoje satove, okreću očima,

and they're all wondering at some point or another,

i svatko od njih u jednom trenutku pomisli,

"Man, how long is it going to take to fill up?"

"Čovječe, koliko dugo će trebati da se napuni?"

(Laughter)

(Smijeh)

That's how you know you've baited the hook, right?

Tako znate da su zagrizli mamac, zar ne?

And that question, off this right here, is really fun for me

I to je pitanje meni stvarno zabavno,

because, like the intro,

jer, kao što sam rekao u uvodu,

I teach kids -- because of my inexperience --

ja poučavam klince, zbog vlastitog neiskustva,

I teach the kids that are the most remedial, all right?

ja učim klince koji najviše zaostaju.

And I've got kids who will not join a conversation about math

I imam klince koji se neće uključiti u razgovor o matematici

because someone else has the formula;

jer netko drugi ima formulu,

someone else knows how to work the formula better than me,

netko drugi zna bolje kako doći do formule od mene.

so I won't talk about it.

Tako da neću o tome govoriti.

But here, every student is on a level playing field of intuition.

svi su ravnopravni kad je riječ o intuiciji.

Everyone's filled something up with water before,

Svatko je jednom punio nešto vodom,

so I get kids answering the question, "How long will it take?"

tako dobijem klince da traže odgovor na pitanje, koliko dugo će trajati.

I've got kids who are mathematically and conversationally intimidated

Imam klince koji se boje i matematike i razgovora

joining the conversation.

koji se uključuju u ovaj razgovor.

We put names on the board, attach them to guesses,

Stavljamo imena na ploču, vežemo ih uz nagađanja,

and kids have bought in here.

i klinci su ovdje uključeni.

And then we follow the process I've described.

I onda pratimo proces koji sam opisao.

And the best part here, or one of the better parts

I najbolja stvar u tome, ili jedna od boljih

is that we don't get our answer from the answer key

je da ne dođemo do rezultata metodom

in the back of the teacher's edition.

koja je opisana u priručniku za nastavnike.

We, instead, just watch the end of the movie.

Mi, umjesto toga, samo pogledamo kraj filma.

(Laughter)

(Smijeh)

And that's terrifying,

I to je zastrašujuće, zar ne?

because the theoretical models that always work out

Jer teoretski modeli koji uvijek rade

in the answer key in the back of a teacher's edition,

u priručniku za nastavnika

that's great, but

to je sjajno, ali

it's scary to talk about sources of error

je zastrašujuće govoriti o uzrocima grešaka

when the theoretical does not match up with the practical.

kada se teoretski ne podudara s praktičnim.

But those conversations have been so valuable,

Ali ti su razgovori toliko dragocjeni,

among the most valuable.

među najvrednijim stvarima.

So I'm here to report some really fun games

Ovdje sam došao kako bih vas izvjestio o nekim zabavnim poboljšanjima

with students who come pre-installed

kod učenika koji su došli predinstalirani

with these viruses day one of the class.

s tim virusima prvog dana u razredu.

These are the kids who now, one semester in,

To su djeca kojoj sada, nakon jednog semestra,

I can put something on the board,

mogu staviti nešto na ploču,

totally new, totally foreign,

skroz novo i potpuno nepoznato,

and they'll have a conversation about it for three or four minutes more

i oni razgovarati o tome 3-4 minute dulje

than they would have at the start of the year,

nego što bi o tome raspravljali na početku godine,

which is just so fun.

što je tako zabavno.

We're no longer averse to word problems,

Više nismo neskloni problemskim zadacima

because we've redefined what a word problem is.

jer smo redefinirali što znače ti problemski zadaci.

We're no longer intimidated by math,

Više nismo zaplašeni matematikom,

because we're slowly redefining what math is.

jer polagano redefiniramo što predstavlja matematika.

This has been a lot of fun.

To je bilo tako zabavno.

I encourage math teachers I talk to to use multimedia,

Želim ohrabriti nastavnike matematike kojima se obraćam, da koriste multimediju,

because it brings the real world into your classroom

jer ona donosi stvarni svijet u učionice

in high resolution and full color;

u visokoj rezoluciji i punoj boji,

to encourage student intuition for that level playing field;

ona ohrabruje učenikovu intuiciju

to ask the shortest question you possibly can

da postavlja najkraća moguća pitanja

and let those more specific questions come out in conversation;

i pusti da se ona specifična pitanja iznjedre u razgovoru,

to let students build the problem,

da učenici sami izgrade problem

because Einstein said so;

jer je Einstein tako rekao,

and to finally, in total, just be less helpful,

i da, na kraju, budemo manje pri ruci

because the textbook is helping you in all the wrong ways:

jer vam udžbenik pomaže na krivi način.

It's buying you out of your obligation,

Oduzima vam obvezu

for patient problem solving and math reasoning, to be less helpful.

strpljivog rješavanja problema i matematičkog promišljanja.

And why this is an amazing time to be a math teacher right now

Zašto je ovo zadivljujuće vrijeme za nastavnike matematike

is because we have the tools to create

jer u džepu imamo alat za kreiranje

this high-quality curriculum in our front pocket.

tog jako kvalitetnog nastavnog programa.

It's ubiquitous and fairly cheap,

Sveprisutan je i prilično jeftin.

and the tools to distribute it

A alati za njegovu distribuciju

freely under open licenses

bez plaćanja, pod otvorenom licencom

has also never been cheaper or more ubiquitous.

također nikada nisu bili jeftiniji i lakše dostupni.

I put a video series on my blog not so long ago

Stavio sam seriju videa na moj blog nedavno

and it got 6,000 views in two weeks.

i 6000 ljudi je to došlo pogledati u samo dva tjedna.

I get emails still from teachers in countries I've never visited

Još uvijek dobijam e-mailove od nastavnika iz zemalja u kojima nikad nisam bio

saying, "Wow, yeah. We had a good conversation about that.

koji govore. "Vau, da. Imali smo dobar razgovor o tome.

Oh, and by the way, here's how I made your stuff better,"

I evo kako ja unapređujem ono što vi radite."

which, wow.

Što je sjajno.

I put this problem on my blog recently:

Ovaj sam zadatak nedavno stavio na svoj blog.

In a grocery store, which line do you get into,

U dućanu, u koji red ćete stati?

the one that has one cart and 19 items

Onaj s jednim kolicima u kojima se nalazi 19 predmeta,

or the line with four carts and three, five, two and one items.

ili u onaj s četvero kolica s po tri, pet, dva i jednim proizvodom?

And the linear modeling involved in that was some good stuff for my classroom,

Linearno modeliranje koje je potrebno za ovaj zadatak je sjajna stvar za moja predavanja

but it eventually got me on "Good Morning America" a few weeks later,

ali sam se ovoga sjetio nekoliko tjedana kasnije dok sam gledao "Dobro jutro Ameriko",

which is just bizarre, right?

što je bizarno, zar ne?

And from all of this, I can only conclude

Iz svega ovoga mogu zaključiti

that people, not just students,

da su ljudi, ne samo učenici,

are really hungry for this.

stvarno gladni ovoga.

Math makes sense of the world.

Matematika nas uči kako shvatiti svijet.

Math is the vocabulary

Matematika je riječnik

for your own intuition.

za vašu vlastitu intuiciju.

So I just really encourage you, whatever your stake is in education --

Dakle ja vas samo ohrabrujem, kakvu god ulogu u obrazovanju imali

whether you're a student, parent, teacher, policy maker, whatever --

jeste li učenici, roditelji, nastavnici, političari, štogod,

insist on better math curriculum.

inzistirajte na boljem programu matematike.

We need more patient problem solvers. Thank you. (Applause)

Treba nam više strpljivih rješavača problema. Hvala vam.