Math class needs a makeover | Dan Meyer

Prevodilac: Elizabeta Petrovic Lektor: Sandra Gojic

Can I ask you to please recall a time

Mogu li da vas zamolim da se setite vremena

when you really loved something --

kada ste stvarno voleli nešto,

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

film, album, pesmu ili knjigu,

and you recommended it wholeheartedly

i preporučili ste to širokog srca

to someone you also really liked,

nekome ko vam se stvarno sviđao,

and you anticipated that reaction, you waited for it,

i očekivali ste reakciju, čekali ste je,

and it came back, and the person hated it?

i došla je, i ta osoba je to mrzela.

So, by way of introduction,

Pa, čisto zbog uvoda,

that is the exact same state

to je tačno stanje

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

u kojem provodim svaki radni dan poslednjih 6 godina.

I teach high school math.

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 neće, ali je obavezno po zakonu da ga kupi.

I mean, it's just a losing proposition.

Mislim, to je unapred izgubljen slučaj.

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

Postoji koristan stereotip o učenicima koje viđam,

a useful stereotype about you all.

koristan stereotip o svima vama.

I could give you guys

Mogu da vam dam

an algebra-two final exam,

finalni test iz algebre 2,

and I would expect no higher

i očekivao bih prolaznost manju od 25%.

than a 25 percent pass rate.

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

I oba podatka govore manje o vama ili mojim učenicima

than they do about what we call math education

nego što govore o matematičkom obrazovanju

in the U.S. today.

danas u SAD-u.

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

Na početku, hteo bih da podelim matematiku na 2 kategorije.

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

Jedna je računanje. Ovo je stvar koju ste zaboravili.

For example, factoring quadratics with

Na primer, faktorizacija kvadratnih jednačina

leading coefficients greater than one.

glavnim koeficijentom većim od 1.

This stuff is also really easy to relearn,

Lako je opet naučiti ovu stvar,

provided you have a really strong grounding

ako imate jako dobru osnovu

in reasoning. Math reasoning --

u razumevanju, matematičkom razumevanju.

we'll call it the application

Zvaćemo ga primena

of math processes to the world around us --

matematičkih procesa na svet oko nas.

this is hard to teach.

Ovo je teško predavati.

This is what we would love students to retain,

Ovo bismo hteli da učenici zadrže,

even if they don't go into mathematical fields.

čak iako ne nastave sa matematikom.

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

Ovo je takođe nešto, s obzirom na način na koji predajemo u SAD-u,

all but ensures they won't retain it.

što sigurno neće zapamtiti.

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

Dakle, reći ću vam zašto je to tako,

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

zašto je to tolika nesreća za društvo, šta možemo da uradimo,

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

i, da završim, zašto je ovo neverovatno vreme

to be a math teacher.

da se bude profesor matematike.

So first, five symptoms

Prvo, 5 simptoma

that you're doing math reasoning wrong

da loše predate matematiku u vašem odeljenju.

in your classroom.

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

Jedan je manjak inicijative; vaši učenici nemaju volje.

You finish your lecture block

Završite predavanje

and immediately you have five hands going up

i odmah imate 5 ruku koje se dižu

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

i pitaju da opet objasnite celo predavanje.

Students lack perseverance.

Učenicima nedostaje istrajnost.

They lack retention; you find yourself

Imaju teškoće sa pamćenjem; i nađete se

re-explaining concepts three months later, wholesale.

kako opet objašnjavate isti koncept 3 meseca kasnije.

There's an aversion to word problems,

Postoji odbojnost prema tekstualnim problemima,

which describes 99 percent of my students.

a to navodi 99 posto mojih učenika.

And then the other one percent

I onda onih jedan odsto

is eagerly looking for the formula

jedva čekaju formulu

to apply in that situation.

da primene u toj situaciji.

This is really destructive.

To je destruktivno.

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

Dejvid Milč, stvaralac "Deadwood"-a i još nekih neverovatnih TV serija,

has a really good description for this.

je ovo dobro opisao.

He swore off creating

Odrekao se stvaranja

contemporary drama,

modernih drama,

shows set in the present day,

serija u sadašnjem vremenu,

because he saw that when people fill their mind

jer je video da kada ljudi pune svoje glave

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

sa 4 sata dnevno, recimo, "Dva i po muškarca", bez uvrede,

it shapes the neural pathways, he said,

to oblikuje neuronske putanje, kako on kaže,

in such a way that they expect simple problems.

na takav način da oni očekuju jednostavne probleme.

He called it, "an impatience with irresolution."

On to zove "nestrpljivost usled nemanja rešenja".

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

Nestrpljivi ste sa stvarima koje se ne rešavaju brzo.

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

Očekujete probleme kao iz sitkoma koji se reše za 22 minuta,

three commercial breaks and a laugh track.

3 bloka reklama i smehom.

And I'll put it to all of you,

I svima ću reći,

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

iako znate, nijedan problem vredan rešavanja nije jednostavan.

I am very concerned about this

I ovo me veoma brine,

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

jer ću se penzionisati u svetu u kojem će moji učenici vladati.

I'm doing bad things

Radim loše stvari

to my own future and well-being

mojoj budućnosti i dobrobiti

when I teach this way.

kada predajem ovako.

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

Ovde sam da vam kažem da način na koji naši udžbenici,

mass-adopted textbooks -- teach math reasoning

naročito opšteprihvaćeni, uče matematičko razmišljanje

and patient problem solving,

i strpljivo rešavanje problema,

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

funkcionalno jednako gledanju "Dva i po muškarca" i mislite da je to to.

(Laughter)

(Smeh)

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

Ozbiljno, evo ga primer iz knjige iz fizike.

It applies equally to math.

Isto važi i za matematiku.

Notice, first of all here,

Pre svega, primetićete

that you have exactly three pieces of information there,

da su ovde date tačno 3 informacije,

each of which will figure into a formula

svaka od njih će se uvrstiti u formulu

somewhere, eventually,

negde, na kraju,

which the student will then compute.

koju će učenik onda da računa.

I believe in real life.

Ja verujem u stvaran život.

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

I pitajte se, koji ste to problem rešili, ikada,

that was worth solving

koji je bio vredan rešavanja,

where you knew all of the given information in advance;

gde ste sve informacije znali unapred,

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

ili gde niste imali višak informacija koje ste morali da filtrirate,

or you didn't have sufficient information

ili gde niste imali dovoljno informacija

and had to go find some.

i morali ste da pronađete još.

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

Siguran sam da nijedan problem vredan rešavanja nije takav.

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

I knjiga, mislim, zna kako je teško učenicima.

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

Jer, gledajte ovo, ovo je set zadataka.

When it comes time to do the actual problem set,

Kada dođe vreme da se stvarno reši problem,

we have problems like this right here

imamo probleme kao ovaj ovde

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

gde samo zamenjujemo brojeve i malo podešavamo kontekst.

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

I ako učenik i dalje ne prepoznaje šablon,

it helpfully explains to you

uspešno vam objašnjava

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

kom primeru problema možete da se vratite da nađete formulu.

You could literally, I mean this,

I bukvalno možete, da uradite to,

pass this particular unit without knowing any physics,

da prođete ovaj deo bez znanja fizike,

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

samo da znate kako da dekodirate knjigu. To je šteta.

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

Tako mogu malo tačnije da dijagnostikujem problem u matematici.

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

Evo jednog sjajnog primera koji mi se dopada.

It's about defining steepness and slope

Radi se definisanju strmine i spusta

using a ski lift.

koristeći ski prevoz.

But what you have here is actually four separate layers,

Ali ovde imate 4 odvojena sloja.

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

I radoznao sam ko od vas može da ih vidi,

and, particularly, how when they're compressed together

i, naročito, kako su spojeni zajedno

and presented to the student all at once,

i predstavljeni učeniku odjednom,

how that creates this impatient problem solving.

kako to stvara nestrpljivo rešavanje problema.

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

Definisaću ih ovde. Imate vizuelni prikaz.

You also have the mathematical structure,

Imate takođe matematičku strukturu,

talking about grids, measurements, labels,

govorim o mreži , merenjima, oznakama,

points, axes, that sort of thing.

tačkama, osama, i takvim stvarima.

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

Imate potkorake, koji svi vode onome o čemu hoćemo da govorimo,

which section is the steepest.

koji deo je najstrmlji.

So I hope you can see.

Nadam se da možete da vidite.

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

Stavrno se nadam da vidite da mi ovde

is taking a compelling question, a compelling answer,

uzimamo zanimljivo pitanje, zanimljiv odgovor,

but we're paving a smooth, straight path

i od toga pravimo gladak, prav put

from one to the other

od jednog do drugog,

and congratulating our students for how well

i čestitamo našim učenicima što mogu

they can step over the small cracks in the way.

dobro da savladaju male pukotine na putu.

That's all we're doing here.

To je sve što radimo ovde.

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

Hoću da vam pokažem da ako možemo da razdvojimo ove na različite načine

and build them up with students,

i da ih izgradimo sa učenicima,

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

možemo da imamo sve što tražimo što se tiče strpljivog rešavanja problema.

So right here I start with the visual,

and I immediately ask the question:

Počeću slikovito,

i odmah pitati:

Which section is the steepest?

koji deo je najstrmiji?

And this starts conversation

I ovo otvara razgovor

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

jer je vizuelizacija napravljena tako da možete da branite 2 odgovora.

So you get people arguing against each other,

Tako da imate ljude koji se međusobno raspravljaju,

friend versus friend,

prijatelj protiv prijatelja,

in pairs, journaling, whatever.

u parovima, kao novinari, kako god.

And then eventually we realize

I onda na kraju shvatimo

it's getting annoying to talk about

da nervira pričanje

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

o skijašu dole levo na ekranu

or the skier just above the mid line.

ili skijašu iznad srednje linije.

And we realize how great would it be

I shvatimo kako bi bilo super

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

da samo imamo oznake A, B, C i D

to talk about them more easily.

da bismo o njima lakše pričali.

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

I dok počinjemo da definišemo pojam strmine,

we realize it would be nice to have some measurements

shvatimo da bi bilo dobro da imamo neke mere

to really narrow it down, specifically what that means.

da suzimo, specifikujemo značenje.

And then and only then,

I tada i samo tada,

we throw down that mathematical structure.

ubacimo matematčku 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

I u tom trenutku, pokazaću vam da je 9 od 10 odeljenja

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

dobro na zadatku spust-strmina.

But if you need to,

Ali ako je potrebno,

your students can then develop those substeps together.

vaši učenici mogu da razviju te potkorake zajedno.

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

Da li, ljudi, vidite, kako ovo, ovde, u poređenju s tim -

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

koje od njih stvara strpljivo rešavanje problema, matematčko razumevanje?

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

Bilo mi je očigledno u mojoj praksi.

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

I na trenutak predajem pozornicu Ajnštajnu,

who, I believe, has paid his dues.

koji, je verujem, platio svoj ceh.

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

On je pričao kako je formulacija problema neverovatno bitna,

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

i ipak u mojoj praksi, ovde u SAD-u,

we just give problems to students;

samo dajemo probleme učenicima,

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

ne uključujemo ih u formulaciju problema.

So 90 percent of what I do

Tako da 90% onoga što radim

with my five hours of prep time per week

sa mojih 5 sati pripremanja nedeljno

is to take fairly compelling elements

je da uzmem nesalomive elemente

of problems like this from my textbook

problema kao ovaj iz knjige

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

i da ih opet izgradim da podržavaju

matematičko razumevanje i strpljivo rešavanje problema.

And here's how it works.

I evo kako to radi.

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

Volim ovo pitanje. U pitanju je sud za vodu.

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

Pitanje je: koliko vremena je potrebno da se napuni? Ok?

First things first, we eliminate all the substeps.

Prvo, eliminišemo sve potkorake.

Students have to develop those,

Učenici moraju da ih razviju.

they have to formulate those.

Moraju da ih formulišu.

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

I onda da primete da su tu napisane sve informacije koje su im potrebne.

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

Ništa ne odvlači pažnju, tako da gubimo to.

Students need to decide, "All right, well,

Učenici treba da odluče, u redu, dobro, da li je visina bitna?

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

Da li je veličina bitna?

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

Da li je boja ventila bitna? Šta je bitno ovde?

Such an underrepresented question in math curriculum.

Toliko nedovoljno predstavljeno pitanje u programu matematike.

So now we have a water tank.

Tako da sada imamo sud za vodu.

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

Koliko vam treba vremena da ga napunite, i to je to.

And because this is the 21st century

I pošto je ovo 21. vek,

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

voleli bismo sa pričamo o realnom svetu u njegovim uslovima,

not in terms of line art or clip art

ne u uslovima linija i slika

that you so often see in textbooks,

koje često vidite u knjigama,

we go out and we take a picture of it.

mi izađemo i slikamo.

So now we have the real deal.

I sada imamo pravu stvar.

How long will it take it to fill it up?

Koliko treba da se napuni?

And then even better is we take a video,

I još bolje, ako uzmemo video,

a video of someone filling it up.

video nekog ko ga puni.

And it's filling up slowly, agonizingly slowly.

I polako se puni, sporo do agonije.

It's tedious.

Dosadno je.

Students are looking at their watches, rolling their eyes,

Učenici gledaju u svoje satove, prevrću očima,

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

i pitaju se u nekom trenutku:

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

"Čoveče, koliko mu vremena treba da se napuni?"

(Laughter)

(Smeh)

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

Tako znate da su se upecali, da.

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

I to pitanje, ovo ovde, mi je zabavno,

because, like the intro,

jer, kao što rekoh,

I teach kids -- because of my inexperience --

učim decu, jer zbog mog neiskustva,

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

učim decu da se ovo najviše popravlja.

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

I imam decu koja neće da se pridruže razgovoru o matematici

because someone else has the formula;

jer neko drugi ima formulu,

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

neko drugi zna da koristi formulu bolje od mene.

so I won't talk about it.

Tako da neću da pričam o tome.

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

Ali ovde, svi su na istom nivou gde se igra intuicijom.

Everyone's filled something up with water before,

Svako je nekad nešto napunio vodom,

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

tako da nateram decu da odgovore, koliko treba da se napuni.

I've got kids who are mathematically and conversationally intimidated

Imam decu koja su matematički i konverzaciono zastrašena

joining the conversation.

da se pridruže razgovoru.

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

Stavljamo imena na tablu, povezujemo ih sa nagađanjima,

and kids have bought in here.

i deca su to prihvatila.

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 najbolji deo ovde, ili jedan od najboljih delova

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

je taj da ne dobijamo naš odgovor iz rešenja

in the back of the teacher's edition.

na kraju profesorovog priručnika..

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

Mi, ustvari, samo gledamo kraj filma.

(Laughter)

(Smeh)

And that's terrifying,

I to je zastrašujuće.

because the theoretical models that always work out

Jer teoretski modeli koji uvek rade

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

u rešenjima na kraju knjige,

that's great, but

oni su super,

it's scary to talk about sources of error

ali strašno je pričati o izvorima grešaka

when the theoretical does not match up with the practical.

kada se teorija ne poklapa sa praktičnim.

But those conversations have been so valuable,

Jer su ti razgovori bili toliko vredni,

among the most valuable.

među najvrednijima.

So I'm here to report some really fun games

Tako da sam ovde da izvestim o zabavnim dobicima

with students who come pre-installed

sa učenicima koju su došli prethodno instalirani sa ovim virusima,

with these viruses day one of the class.

prvog dana predavanja.

These are the kids who now, one semester in,

Ovo su deca koja sada, posle jednog polugodišta,

I can put something on the board,

ako stavim nešto na tablu,

totally new, totally foreign,

totalno novo, strano,

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

oni će da pričaju o tome 3-4 minuta više

than they would have at the start of the year,

nego na početku godine,

which is just so fun.

što je baš zabavno.

We're no longer averse to word problems,

Više nemamo averziju prema tekstualnim problemima,

because we've redefined what a word problem is.

jer smo redefinisali šta je tekstualni problem.

We're no longer intimidated by math,

Nismo više zastrašeni matematikom,

because we're slowly redefining what math is.

jer polako redefinišemo šta je matematika.

This has been a lot of fun.

Ovo je bilo baš zabavno.

I encourage math teachers I talk to to use multimedia,

Ohrabrujem nastavnike matematike sa kojima pričam da koriste multimediju,

because it brings the real world into your classroom

jer to uvodi realni život u vašu učionicu

in high resolution and full color;

sa visokom rezolucijom i bojom,

to encourage student intuition for that level playing field;

da podstičemo intuiciju učenika da dođe na taj nivo,

to ask the shortest question you possibly can

da pitate najkraća pitanja koja možete

and let those more specific questions come out in conversation;

i da dopustite da se ta specifičnija pitanja jave u razgovoru,

to let students build the problem,

da dozvolite učenicima da sagrade problem,

because Einstein said so;

jer je Ajnštajn rekao tako,

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

i na kraju, sve u svemu, da manje pomažete,

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

jer knjiga vam pomaže na pogrešne načine.

It's buying you out of your obligation,

Ona vas vadi iz obaveze da strpljivo

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

rešavate problem i matematički razmišljate, da manje pomažete.

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

I zato je ovo neverovatno vreme da budete nastavnik matematike

is because we have the tools to create

jer imamo alate da stvorimo

this high-quality curriculum in our front pocket.

program visokog kvaliteta u našem prednjem džepu.

It's ubiquitous and fairly cheap,

To je sveprisutno i dosta jeftino.

and the tools to distribute it

I alat za njihovu slobodnu distribuciju

freely under open licenses

sa otvorenim licencama

has also never been cheaper or more ubiquitous.

takođe nikada nije bilo jeftinije i sveprisutnije.

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

Postavio sam seriju klipova na mom blogu ne tako davno,

and it got 6,000 views in two weeks.

i imali su 6 000 pregleda za 2 nedelje.

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

I dalje dobijam mailove od profesora iz zemalja koje nikada nisam posetio

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

gde kažu: "Opa, da. Imali smo dobar razgovor o tome.

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

Oh, i usput, evo kako sam popravio tvoju stvar",

which, wow.

što je sjajno.

I put this problem on my blog recently:

Postavio sam ovaj problem skoro na mom blogu.

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

U prodavnici, u koji red da stanete,

the one that has one cart and 19 items

onaj koji ima jedna kolica i 19 stvari,

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

ili onaj sa 4 kolica i 3, 5, 2 i jednom stvari.

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

I linearno modeliranje ovde uključeno, bila je dobra stvar za moje učenike,

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

ali me je na kraju dovelo u "Dobro jutro Ameriko"

which is just bizarre, right?

nekoliko nedelja kasnije, što je bizarno.

And from all of this, I can only conclude

I iz svega ovoga mogu samo da zaključim

that people, not just students,

da su ljudi, ne samo učenici,

are really hungry for this.

željni ovoga.

Math makes sense of the world.

Matematika daje smisao svetu.

Math is the vocabulary

Matematika je rečnik

for your own intuition.

vaše intuicije.

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

Tako da vas podržavam, kako god da učestvujete u obrazovanju,

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

bilo da ste učenik, roditelj, nastavnik, stvarate pravila, svejedno,

insist on better math curriculum.

insistirajte na boljem matematičkom programu.

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

Treba nam više strpljivih rešavača problema. Hvala vam.