Can I ask you to please recall a time when you really loved something -- a movie, an album, a song or a book -- and you recommended it wholeheartedly to someone you also really liked, and you anticipated that reaction, you waited for it, and it came back, and the person hated it? So, by way of introduction, that is the exact same state in which I spent every working day of the last six years. (Laughter) I teach high school math. I sell a product to a market that doesn't want it, but is forced by law to buy it. I mean, it's just a losing proposition.
Mogu li vas zamoliti da se prisjetite vremena kada ste uistinu nešto voljeli, film, album, pjesmu ili knjigu, i preporučili ste to iskreno nekome do koga vam je stalo, i očekivali ste tu reakciju, čekali ste na nju, i ona je došla; osoba je mrzila vašu preporuku. Čisto uvoda radi ovo je upravo stanje u kojem provodim gotovo svaki dan u posljednjih šest godina. Naime, predajem matematiku u srednjoj školi. Prodajem proizvod tržištu koje ga ne želi, ali ga je zakonom prisiljeno kupiti. To je unaprijed izgubljen slučaj.
So there's a useful stereotype about students that I see, a useful stereotype about you all. I could give you guys an algebra-two final exam, and I would expect no higher than a 25 percent pass rate. And both of these facts say less about you or my students than they do about what we call math education in the U.S. today.
Postoji koristan stereotip o učenicima koji ja vidim, koristan stereotip o svima vama. Mogao bih vam dati završni ispit iz algebre dva i ne bih očekivao veću prolaznost od 25%. Obje ove činjenice govore manje o vama ili mojim učenicima nego o onome što nazivamo matematičko obrazovanje u današnjoj Americi.
To start with, I'd like to break math down into two categories. One is computation; this is the stuff you've forgotten. For example, factoring quadratics with leading coefficients greater than one. This stuff is also really easy to relearn, provided you have a really strong grounding in reasoning. Math reasoning -- we'll call it the application of math processes to the world around us -- this is hard to teach. This is what we would love students to retain, even if they don't go into mathematical fields. This is also something that, the way we teach it in the U.S. all but ensures they won't retain it. So, I'd like to talk about why that is, why that's such a calamity for society, what we can do about it and, to close with, why this is an amazing time to be a math teacher.
Krenuo bih s time da podijelim matematiku na dvije kategorije. Jedna je računanje. To su stvari koje ste zaboravili. Na primjer, rastavljanje kvadrata s vodećim koeficijentima koji su veći od jedan. Te je stvari također jako jednostavno ponovno naučiti, pod uvjetom da imate stvarno jako razvijeno racionalno razmišljanje, matematičko promišljanje. Nazvati ćemo to primjenom matematičkih procesa na svijet oko nas. To je teško za naučiti nekoga. To je ono što bismo voljeli da naši učenici dobiju i zadrže, čak i ako ne nastave obrazovanje u matematičkom području. To je također nešto, s obzirom na način na koji poučavamo u Americi što oni vrlo vjerojatno neće zadržati. Tako da ću ja govoriti o tome zašto je to tako, zašto je to nesreća za društvo, što možemo po tom pitanju učiniti, i, za kraj, zašto je ovo tako nevjerojatno vrijeme biti nastavnikom matematike.
So first, five symptoms that you're doing math reasoning wrong in your classroom. One is a lack of initiative; your students don't self-start. You finish your lecture block and immediately you have five hands going up asking you to re-explain the entire thing at their desks. Students lack perseverance. They lack retention; you find yourself re-explaining concepts three months later, wholesale. There's an aversion to word problems, which describes 99 percent of my students. And then the other one percent is eagerly looking for the formula to apply in that situation. This is really destructive.
Prije svega, pet simptoma koji pokazuju da krivo poučavate matematiku u svom razredu. Prvi je manjak inicijative; vaši studenti ne započinju sami. Nakon što završite sa svojim blokom predavanja imate po pravilu barem pet ruku u zraku koje od vas traže da ponovno objasnite cijelu stvar. Učenicima nedostaje upornosti. Njima nedostaje sposobnosti da zapamte; često smo u situaciji da moram ponovno objašnjavati koncepte tri mjeseca kasnije. Ne vole problemske zadatke, one s riječima a takvih je 99% mojih učenika. A onih 1% nestrpljivo traži formulu koja se može primijeniti na zadatak. To je stvarno destruktivno.
David Milch, creator of "Deadwood" and other amazing TV shows, has a really good description for this. He swore off creating contemporary drama, shows set in the present day, because he saw that when people fill their mind with four hours a day of, for example, "Two and a Half Men," no disrespect, it shapes the neural pathways, he said, in such a way that they expect simple problems. He called it, "an impatience with irresolution." You're impatient with things that don't resolve quickly. You expect sitcom-sized problems that wrap up in 22 minutes, three commercial breaks and a laugh track. And I'll put it to all of you, what you already know, that no problem worth solving is that simple. I am very concerned about this because I'm going to retire in a world that my students will run. I'm doing bad things to my own future and well-being when I teach this way. I'm here to tell you that the way our textbooks -- particularly mass-adopted textbooks -- teach math reasoning and patient problem solving, it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
David Milch, autor serije "Deadwood" i drugih čudesnih TV emisija, ima stvarno dobar opis ovoga. Zarekao se da će stvoriti suvremenu dramu, koja će se prikazivati u sadašnjosti, jer je shvatio da kada ljudi ispune um s četiri sada dnevno, na primjer serije "Dva i pol muškarca", uz dužno poštovanje, oblikuju im se veze u mozgu, kaže, na takav način da očekuju samo jednostavne probleme. On je to nazvao, "nestrpljivost za rješenje". Nestrpljivi ste sa stvarima koje se ne razriješe brzo. Očekujete jednostavne probleme kao u serijama koje stanu u 22 minute, uključujuće tri pauze za reklame i vrijeme za smijeh. I neću vam otkriti novost ako vam kažem da niti jedan problem vrijedan rješavanja nije jednostavan. Jako sam zabrinut zbog svega toga, jer ja ću otići u mirovinu u svijetu koji će voditi moji učenici. Činim loše stvari vlastitoj budućnosti i blagostanju kada poučavam na taj način. Ovdje sam da vam kažem kako je način na koji naši udžbenici, posebno općeprihvaćeni udžbenici, podučavaju matematičko promišljanje i strpljivo rješavanje problema, 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. It applies equally to math. Notice, first of all here, that you have exactly three pieces of information there, each of which will figure into a formula somewhere, eventually, which the student will then compute. I believe in real life. And ask yourself, what problem have you solved, ever, that was worth solving where you knew all of the given information in advance; where you didn't have a surplus of information and you had to filter it out, or you didn't have sufficient information and had to go find some. I'm sure we all agree that no problem worth solving is like that. And the textbook, I think, knows how it's hamstringing students because, watch this, this is the practice problem set. When it comes time to do the actual problem set, we have problems like this right here where we're just swapping out numbers and tweaking the context a little bit. And if the student still doesn't recognize the stamp this was molded from, it helpfully explains to you what sample problem you can return to to find the formula. You could literally, I mean this, pass this particular unit without knowing any physics, just knowing how to decode a textbook. That's a shame.
Za ozbiljno, ovdje je primjer iz udžbenika iz fizike. Jednako je primjenjiv i na matematiku. Primjetiti ćete prije svega kako su dane točno tri informacije ovdje, od kojih će se svaka preobličiti u formulu negdje, na kraju, koju će učenici onda izračunati. Mislim kako u stvarnom životu to ne postoji. Zapitajte se: koji ste problem ikada riješili, a koji je bio vrijedan rješavanja, gdje ste znali sve ključne informacije unaprijed, ili niste imali višak informacija od kojih ste trebali izabrati ključne, ili niste imali manjak informacija i trebali ste ih pronaći negdje. Siguran sam da ćemo se svi složiti kako niti jedan problem vrijedan rješavanja nije takav. Udžbenik, mislim, zna kako sakati učenike. Jer, pogledajte ovo, ovo je praktični zadatak. Kada dođe vrijeme za rješavanje stvarnih zadataka, zadaju se zadaci poput ovoga ovdje gdje samo izmjenimo brojeve i malo promjenimo kontekst. Ako učenik još uvijek ne prepozna po kojoj se šabloni zadatak rješava prijateljski vam objasni u kojem ranijem primjeru možete pronaći pravu formulu za rješenje. Mogli bi doslovno, to stvarno mislim, proći ovo poglavlje bez imalo razumijevanja fizike, samo treba znati kako dešifrirati udžbenik. To je velika šteta.
So I can diagnose the problem a little more specifically in math. Here's a really cool problem. I like this. It's about defining steepness and slope using a ski lift. But what you have here is actually four separate layers, and I'm curious which of you can see the four separate layers and, particularly, how when they're compressed together and presented to the student all at once, how that creates this impatient problem solving. I'll define them here: You have the visual. You also have the mathematical structure, talking about grids, measurements, labels, points, axes, that sort of thing. You have substeps, which all lead to what we really want to talk about: which section is the steepest.
Problem mogu još bolje dijagnosticirati u matematici. Ovdje je stvarno zgodan zadatak. Odnosi se na definiranje kosine i nagiba koristeći ski lift. Ali ono što imate ovdje jesu u stvari četiri odvojene razine. I baš me zanima tko od vas može vidjeti četiri odvojene razine i, posebno, kako to što su pomiješane zajedno i predstavljeni učenicima svi odjednom, kako to uzrokuje taj proces nestrpljivog rješavanja problema. Definirati ću ih ovdje. Imate sliku. Imate i matematičku strukturu govorim o mrežama, mjerama, oznakama, točkama, osima, i tako dalje. Imate međukorake, koji vas vode do onoga o čemu uistinu želimo razgovarati, koja dio skijaške staze je najstrmiji.
So I hope you can see. I really hope you can see how what we're doing here is taking a compelling question, a compelling answer, but we're paving a smooth, straight path from one to the other and congratulating our students for how well they can step over the small cracks in the way. That's all we're doing here. So I want to put to you that if we can separate these in a different way and build them up with students, we can have everything we're looking for in terms of patient problem solving.
Nadam se da možete vidjeti. Stvarno se nadam da možete vidjeti, kako uzmijaći izazovno pitanje, izazovni odgovor mi utiremo glatki, ravni put od jednog prema drugom, i čestitamo našim učenicima koliko dobro mogu savladati male pukotine po tom putu. To je ono što radimo. Tako vas ja želim potaknuti da razmišljamo na drugačiji način i sagradimo taj put zajedno sa učenicima, možemo imati sve ono što želimo s aspekta strpljivog rješavanja zadatka.
So right here I start with the visual, and I immediately ask the question: Which section is the steepest? And this starts conversation because the visual is created in such a way where you can defend two answers. So you get people arguing against each other, friend versus friend, in pairs, journaling, whatever. And then eventually we realize it's getting annoying to talk about the skier in the lower left-hand side of the screen or the skier just above the mid line. And we realize how great would it be if we just had some A, B, C and D labels to talk about them more easily. And then as we start to define what does steepness mean, we realize it would be nice to have some measurements to really narrow it down, specifically what that means. And then and only then, we throw down that mathematical structure. The math serves the conversation, the conversation doesn't serve the math. And at that point, I'll put it to you that nine out of 10 classes are good to go on the whole slope, steepness thing. But if you need to, your students can then develop those substeps together.
Ovdje ću započeti sa skicom i odmah vas upitati: Koji dio staze je najstrmiji? I time ćemo započeti razogovor jer je skica napravljena na takav način da mogu biti dva odgovora. Pa će se ljudi prepirati prijatelji protiv prijatelja, u parovima, bilokako. I onda ćemo na kraju spoznati postaje zamorno nekome govoriti o skijašu u donjoj lijevoj strani ekrana ili o skijašu iznad srednje linije. I shvatimo koliko bi sjajno bilo kada bismo imali samo A, B, C i D oznake o kojima možemo jednostavno razgovarati. A kad počnemo definirati što strmina znači, shvatimo, kako bi bilo dobro imati neku mjeru kako bismo suzili značenje, na specifično značenje. I tek tada, ubacimo matematičku strukturu. Matematika služi razgovoru. Razgovor ne služi matematici. U tom trenutku, uvjeravam vas, 9 od 10 razreda je sposobno razmatrati problem kosine, strmine. Ali ako je potrebno vaši učenici mogu onda razviti te podkorake zajedno.
Do you guys see how this, right here, compared to that -- which one creates that patient problem solving, that math reasoning? It's been obvious in my practice, to me. And I'll yield the floor here for a second to Einstein, who, I believe, has paid his dues. He talked about the formulation of a problem being so incredibly important, and yet in my practice, in the U.S. here, we just give problems to students; we don't involve them in the formulation of the problem.
Vidite li vi kako ovo ovdje, uspoređeno s ovime -- koji od njih uči, dovodi do strpljivog rješavanja problema, matematičkog promišljanja? Prema praksi koju imam meni je odgovor očit. I podijeliti ću podij na sekundu s Einsteinom, koji je, vjerujem, to shvatio iskustvom. On je govorio o tome koliko je iznimno važno formuliranje problema, ipak u mojoj praksi, ovdje u SAD-u, mi jednostavno dzadajemo učenicima zadatke; ne uključujemo ih u formuliranje problema.
So 90 percent of what I do with my five hours of prep time per week is to take fairly compelling elements of problems like this from my textbook and rebuild them in a way that supports math reasoning and patient problem solving. And here's how it works. I like this question. It's about a water tank. The question is: How long will it take you to fill it up? First things first, we eliminate all the substeps. Students have to develop those, they have to formulate those. And then notice that all the information written on there is stuff you'll need. None of it's a distractor, so we lose that. Students need to decide, "All right, well, does the height matter? Does the side of it matter? Does the color of the valve matter? What matters here?" Such an underrepresented question in math curriculum. So now we have a water tank. How long will it take you to fill it up? And that's it.
Tako je 90% onoga što radim u svojih pet sati priprema tjedno jest da uzmem poticajne elemente zadataka poput ovih iz mog udžbenika i preoblikujem ih tako da podržavaju matematičko promišljanje i strpljivo rješavanje problema. I ovdje je prikazano kako to radi. Volim ovakva pitanja. Radi se o spremniku za vodu. Pitanje je: Koliko je vremena potrebno da se napuni? Dobro? Kao prvo, eliminiramo sve pod korake. Učenici ih sami moraju razviti. Oni ih moraju formulirati. I sada možete primjetiti kako su sve informacije napisane ovdje one stvari koje trebate. Niti jedna od njih ne zbunjuje, tako ćemo ih propustiti. Učenici moraju odlučiti, je li visina važna? Je li veličina važna? Je li važna boja ventila? Što je važno ovdje? Ovo je podcjenjeno pitanje u nastavi matematike. Tako imamo spremnik za vodu. Koliko dugo će trebati da se napuni, i to je to.
And because this is the 21st century and we would love to talk about the real world on its own terms, not in terms of line art or clip art that you so often see in textbooks, we go out and we take a picture of it. So now we have the real deal. How long will it take it to fill it up? And then even better is we take a video, a video of someone filling it up. And it's filling up slowly, agonizingly slowly. It's tedious. Students are looking at their watches, rolling their eyes, and they're all wondering at some point or another, "Man, how long is it going to take to fill up?" (Laughter) That's how you know you've baited the hook, right?
I zato jer je ovo 21. stoljeće, i zato jer bismo htjeli raspravljati o stvarnom svijetu kakav jest, a ne kao skica, crtež, što tako često vidimo u udžbenicima, izađemo i fotografiramo ga. Sada imamo zadatak iz stvarnog života. Koliko je potrebno da se napuni? Ili, čak bolje, napravimo video, snimku nekoga tko ga puni. A puni se polako, agonizirajuće polako. To je mučno. Učenici gledaju na svoje satove, okreću očima, i svatko od njih u jednom trenutku pomisli, "Čovječe, koliko dugo će trebati da se napuni?" (Smijeh) Tako znate da su zagrizli mamac, zar ne?
And that question, off this right here, is really fun for me because, like the intro, I teach kids -- because of my inexperience -- I teach the kids that are the most remedial, all right? And I've got kids who will not join a conversation about math because someone else has the formula; someone else knows how to work the formula better than me, so I won't talk about it. But here, every student is on a level playing field of intuition. Everyone's filled something up with water before, so I get kids answering the question, "How long will it take?" I've got kids who are mathematically and conversationally intimidated joining the conversation. We put names on the board, attach them to guesses, and kids have bought in here. And then we follow the process I've described. And the best part here, or one of the better parts is that we don't get our answer from the answer key in the back of the teacher's edition. We, instead, just watch the end of the movie. (Laughter) And that's terrifying, because the theoretical models that always work out in the answer key in the back of a teacher's edition, that's great, but it's scary to talk about sources of error when the theoretical does not match up with the practical. But those conversations have been so valuable, among the most valuable.
I to je pitanje meni stvarno zabavno, jer, kao što sam rekao u uvodu, ja poučavam klince, zbog vlastitog neiskustva, ja učim klince koji najviše zaostaju. I imam klince koji se neće uključiti u razgovor o matematici jer netko drugi ima formulu, netko drugi zna bolje kako doći do formule od mene. Tako da neću o tome govoriti. svi su ravnopravni kad je riječ o intuiciji. Svatko je jednom punio nešto vodom, tako dobijem klince da traže odgovor na pitanje, koliko dugo će trajati. Imam klince koji se boje i matematike i razgovora koji se uključuju u ovaj razgovor. Stavljamo imena na ploču, vežemo ih uz nagađanja, i klinci su ovdje uključeni. I onda pratimo proces koji sam opisao. I najbolja stvar u tome, ili jedna od boljih je da ne dođemo do rezultata metodom koja je opisana u priručniku za nastavnike. Mi, umjesto toga, samo pogledamo kraj filma. (Smijeh) I to je zastrašujuće, zar ne? Jer teoretski modeli koji uvijek rade u priručniku za nastavnika to je sjajno, ali je zastrašujuće govoriti o uzrocima grešaka kada se teoretski ne podudara s praktičnim. Ali ti su razgovori toliko dragocjeni, među najvrednijim stvarima.
So I'm here to report some really fun games with students who come pre-installed with these viruses day one of the class. These are the kids who now, one semester in, I can put something on the board, totally new, totally foreign, and they'll have a conversation about it for three or four minutes more than they would have at the start of the year, which is just so fun. We're no longer averse to word problems, because we've redefined what a word problem is. We're no longer intimidated by math, because we're slowly redefining what math is. This has been a lot of fun.
Ovdje sam došao kako bih vas izvjestio o nekim zabavnim poboljšanjima kod učenika koji su došli predinstalirani s tim virusima prvog dana u razredu. To su djeca kojoj sada, nakon jednog semestra, mogu staviti nešto na ploču, skroz novo i potpuno nepoznato, i oni razgovarati o tome 3-4 minute dulje nego što bi o tome raspravljali na početku godine, što je tako zabavno. Više nismo neskloni problemskim zadacima jer smo redefinirali što znače ti problemski zadaci. Više nismo zaplašeni matematikom, jer polagano redefiniramo što predstavlja matematika. To je bilo tako zabavno.
I encourage math teachers I talk to to use multimedia, because it brings the real world into your classroom in high resolution and full color; to encourage student intuition for that level playing field; to ask the shortest question you possibly can and let those more specific questions come out in conversation; to let students build the problem, because Einstein said so; and to finally, in total, just be less helpful, because the textbook is helping you in all the wrong ways: It's buying you out of your obligation, for patient problem solving and math reasoning, to be less helpful.
Želim ohrabriti nastavnike matematike kojima se obraćam, da koriste multimediju, jer ona donosi stvarni svijet u učionice u visokoj rezoluciji i punoj boji, ona ohrabruje učenikovu intuiciju da postavlja najkraća moguća pitanja i pusti da se ona specifična pitanja iznjedre u razgovoru, da učenici sami izgrade problem jer je Einstein tako rekao, i da, na kraju, budemo manje pri ruci jer vam udžbenik pomaže na krivi način. Oduzima vam obvezu strpljivog rješavanja problema i matematičkog promišljanja.
And why this is an amazing time to be a math teacher right now is because we have the tools to create this high-quality curriculum in our front pocket. It's ubiquitous and fairly cheap, and the tools to distribute it freely under open licenses has also never been cheaper or more ubiquitous. I put a video series on my blog not so long ago and it got 6,000 views in two weeks. I get emails still from teachers in countries I've never visited saying, "Wow, yeah. We had a good conversation about that. Oh, and by the way, here's how I made your stuff better," which, wow. I put this problem on my blog recently: In a grocery store, which line do you get into, the one that has one cart and 19 items or the line with four carts and three, five, two and one items. And the linear modeling involved in that was some good stuff for my classroom, but it eventually got me on "Good Morning America" a few weeks later, which is just bizarre, right?
Zašto je ovo zadivljujuće vrijeme za nastavnike matematike jer u džepu imamo alat za kreiranje tog jako kvalitetnog nastavnog programa. Sveprisutan je i prilično jeftin. A alati za njegovu distribuciju bez plaćanja, pod otvorenom licencom također nikada nisu bili jeftiniji i lakše dostupni. Stavio sam seriju videa na moj blog nedavno i 6000 ljudi je to došlo pogledati u samo dva tjedna. Još uvijek dobijam e-mailove od nastavnika iz zemalja u kojima nikad nisam bio koji govore. "Vau, da. Imali smo dobar razgovor o tome. I evo kako ja unapređujem ono što vi radite." Što je sjajno. Ovaj sam zadatak nedavno stavio na svoj blog. U dućanu, u koji red ćete stati? Onaj s jednim kolicima u kojima se nalazi 19 predmeta, ili u onaj s četvero kolica s po tri, pet, dva i jednim proizvodom? Linearno modeliranje koje je potrebno za ovaj zadatak je sjajna stvar za moja predavanja ali sam se ovoga sjetio nekoliko tjedana kasnije dok sam gledao "Dobro jutro Ameriko", što je bizarno, zar ne?
And from all of this, I can only conclude that people, not just students, are really hungry for this. Math makes sense of the world. Math is the vocabulary for your own intuition. So I just really encourage you, whatever your stake is in education -- whether you're a student, parent, teacher, policy maker, whatever -- insist on better math curriculum. We need more patient problem solvers. Thank you. (Applause)
Iz svega ovoga mogu zaključiti da su ljudi, ne samo učenici, stvarno gladni ovoga. Matematika nas uči kako shvatiti svijet. Matematika je riječnik za vašu vlastitu intuiciju. Dakle ja vas samo ohrabrujem, kakvu god ulogu u obrazovanju imali jeste li učenici, roditelji, nastavnici, političari, štogod, inzistirajte na boljem programu matematike. Treba nam više strpljivih rješavača problema. Hvala vam.