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 da vas zamolim da se setite vremena kada ste stvarno voleli nešto, film, album, pesmu ili knjigu, i preporučili ste to širokog srca nekome ko vam se stvarno sviđao, i očekivali ste reakciju, čekali ste je, i došla je, i ta osoba je to mrzela. Pa, čisto zbog uvoda, to je tačno stanje u kojem provodim svaki radni dan poslednjih 6 godina. Predajem matematiku u srednjoj školi. Prodajem proizvod tržištu koje ga neće, ali je obavezno po zakonu da ga kupi. Mislim, to je unapred 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 koje viđam, koristan stereotip o svima vama. Mogu da vam dam finalni test iz algebre 2, i očekivao bih prolaznost manju od 25%. I oba podatka govore manje o vama ili mojim učenicima nego što govore o matematičkom obrazovanju danas u SAD-u.
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.
Na početku, hteo bih da podelim matematiku na 2 kategorije. Jedna je računanje. Ovo je stvar koju ste zaboravili. Na primer, faktorizacija kvadratnih jednačina glavnim koeficijentom većim od 1. Lako je opet naučiti ovu stvar, ako imate jako dobru osnovu u razumevanju, matematičkom razumevanju. Zvaćemo ga primena matematičkih procesa na svet oko nas. Ovo je teško predavati. Ovo bismo hteli da učenici zadrže, čak iako ne nastave sa matematikom. Ovo je takođe nešto, s obzirom na način na koji predajemo u SAD-u, što sigurno neće zapamtiti. Dakle, reći ću vam zašto je to tako, zašto je to tolika nesreća za društvo, šta možemo da uradimo, i, da završim, zašto je ovo neverovatno vreme da se bude profesor 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.
Prvo, 5 simptoma da loše predate matematiku u vašem odeljenju. Jedan je manjak inicijative; vaši učenici nemaju volje. Završite predavanje i odmah imate 5 ruku koje se dižu i pitaju da opet objasnite celo predavanje. Učenicima nedostaje istrajnost. Imaju teškoće sa pamćenjem; i nađete se kako opet objašnjavate isti koncept 3 meseca kasnije. Postoji odbojnost prema tekstualnim problemima, a to navodi 99 posto mojih učenika. I onda onih jedan odsto jedva čekaju formulu da primene u toj situaciji. To je 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.
Dejvid Milč, stvaralac "Deadwood"-a i još nekih neverovatnih TV serija, je ovo dobro opisao. Odrekao se stvaranja modernih drama, serija u sadašnjem vremenu, jer je video da kada ljudi pune svoje glave sa 4 sata dnevno, recimo, "Dva i po muškarca", bez uvrede, to oblikuje neuronske putanje, kako on kaže, na takav način da oni očekuju jednostavne probleme. On to zove "nestrpljivost usled nemanja rešenja". Nestrpljivi ste sa stvarima koje se ne rešavaju brzo. Očekujete probleme kao iz sitkoma koji se reše za 22 minuta, 3 bloka reklama i smehom. I svima ću reći, iako znate, nijedan problem vredan rešavanja nije jednostavan. I ovo me veoma brine, jer ću se penzionisati u svetu u kojem će moji učenici vladati. Radim loše stvari mojoj budućnosti i dobrobiti kada predajem ovako. Ovde sam da vam kažem da način na koji naši udžbenici, naročito opšteprihvaćeni, uče matematičko razmišljanje i strpljivo rešavanje problema, 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. 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.
Ozbiljno, evo ga primer iz knjige iz fizike. Isto važi i za matematiku. Pre svega, primetićete da su ovde date tačno 3 informacije, svaka od njih će se uvrstiti u formulu negde, na kraju, koju će učenik onda da računa. Ja verujem u stvaran život. I pitajte se, koji ste to problem rešili, ikada, koji je bio vredan rešavanja, gde ste sve informacije znali unapred, ili gde niste imali višak informacija koje ste morali da filtrirate, ili gde niste imali dovoljno informacija i morali ste da pronađete još. Siguran sam da nijedan problem vredan rešavanja nije takav. I knjiga, mislim, zna kako je teško učenicima. Jer, gledajte ovo, ovo je set zadataka. Kada dođe vreme da se stvarno reši problem, imamo probleme kao ovaj ovde gde samo zamenjujemo brojeve i malo podešavamo kontekst. I ako učenik i dalje ne prepoznaje šablon, uspešno vam objašnjava kom primeru problema možete da se vratite da nađete formulu. I bukvalno možete, da uradite to, da prođete ovaj deo bez znanja fizike, samo da znate kako da dekodirate knjigu. To je 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.
Tako mogu malo tačnije da dijagnostikujem problem u matematici. Evo jednog sjajnog primera koji mi se dopada. Radi se definisanju strmine i spusta koristeći ski prevoz. Ali ovde imate 4 odvojena sloja. I radoznao sam ko od vas može da ih vidi, i, naročito, kako su spojeni zajedno i predstavljeni učeniku odjednom, kako to stvara nestrpljivo rešavanje problema. Definisaću ih ovde. Imate vizuelni prikaz. Imate takođe matematičku strukturu, govorim o mreži , merenjima, oznakama, tačkama, osama, i takvim stvarima. Imate potkorake, koji svi vode onome o čemu hoćemo da govorimo, koji deo je najstrmlji.
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 da vidite. Stavrno se nadam da vidite da mi ovde uzimamo zanimljivo pitanje, zanimljiv odgovor, i od toga pravimo gladak, prav put od jednog do drugog, i čestitamo našim učenicima što mogu dobro da savladaju male pukotine na putu. To je sve što radimo ovde. Hoću da vam pokažem da ako možemo da razdvojimo ove na različite načine i da ih izgradimo sa učenicima, 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: 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.
Počeću slikovito, i odmah pitati: koji deo je najstrmiji? I ovo otvara razgovor jer je vizuelizacija napravljena tako da možete da branite 2 odgovora. Tako da imate ljude koji se međusobno raspravljaju, prijatelj protiv prijatelja, u parovima, kao novinari, kako god. I onda na kraju shvatimo da nervira pričanje o skijašu dole levo na ekranu ili skijašu iznad srednje linije. I shvatimo kako bi bilo super da samo imamo oznake A, B, C i D da bismo o njima lakše pričali. I dok počinjemo da definišemo pojam strmine, shvatimo da bi bilo dobro da imamo neke mere da suzimo, specifikujemo značenje. I tada i samo tada, ubacimo matematčku strukturu. Matematika služi razgovoru. Razgovor ne služi matematici. I u tom trenutku, pokazaću vam da je 9 od 10 odeljenja dobro na zadatku spust-strmina. Ali ako je potrebno, vaši učenici mogu da razviju te potkorake 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.
Da li, ljudi, vidite, kako ovo, ovde, u poređenju s tim - koje od njih stvara strpljivo rešavanje problema, matematčko razumevanje? Bilo mi je očigledno u mojoj praksi. I na trenutak predajem pozornicu Ajnštajnu, koji, je verujem, platio svoj ceh. On je pričao kako je formulacija problema neverovatno bitna, i ipak u mojoj praksi, ovde u SAD-u, samo dajemo probleme učenicima, ne uključujemo ih u formulaciju 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 da 90% onoga što radim sa mojih 5 sati pripremanja nedeljno je da uzmem nesalomive elemente problema kao ovaj iz knjige i da ih opet izgradim da podržavaju matematičko razumevanje i strpljivo rešavanje problema. I evo kako to radi. Volim ovo pitanje. U pitanju je sud za vodu. Pitanje je: koliko vremena je potrebno da se napuni? Ok? Prvo, eliminišemo sve potkorake. Učenici moraju da ih razviju. Moraju da ih formulišu. I onda da primete da su tu napisane sve informacije koje su im potrebne. Ništa ne odvlači pažnju, tako da gubimo to. Učenici treba da odluče, u redu, dobro, da li je visina bitna? Da li je veličina bitna? Da li je boja ventila bitna? Šta je bitno ovde? Toliko nedovoljno predstavljeno pitanje u programu matematike. Tako da sada imamo sud za vodu.
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?
Koliko vam treba vremena da ga napunite, i to je to. I pošto je ovo 21. vek, voleli bismo sa pričamo o realnom svetu u njegovim uslovima, ne u uslovima linija i slika koje često vidite u knjigama, mi izađemo i slikamo. I sada imamo pravu stvar. Koliko treba da se napuni? I još bolje, ako uzmemo video, video nekog ko ga puni. I polako se puni, sporo do agonije. Dosadno je. Učenici gledaju u svoje satove, prevrću očima, i pitaju se u nekom trenutku: "Čoveče, koliko mu vremena treba da se napuni?" (Smeh) Tako znate da su se upecali, da.
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 pitanje, ovo ovde, mi je zabavno, jer, kao što rekoh, učim decu, jer zbog mog neiskustva, učim decu da se ovo najviše popravlja. I imam decu koja neće da se pridruže razgovoru o matematici jer neko drugi ima formulu, neko drugi zna da koristi formulu bolje od mene. Tako da neću da pričam o tome. Ali ovde, svi su na istom nivou gde se igra intuicijom. Svako je nekad nešto napunio vodom, tako da nateram decu da odgovore, koliko treba da se napuni. Imam decu koja su matematički i konverzaciono zastrašena da se pridruže razgovoru. Stavljamo imena na tablu, povezujemo ih sa nagađanjima, i deca su to prihvatila. I onda pratimo proces koji sam opisao. I najbolji deo ovde, ili jedan od najboljih delova je taj da ne dobijamo naš odgovor iz rešenja na kraju profesorovog priručnika.. Mi, ustvari, samo gledamo kraj filma. (Smeh) I to je zastrašujuće. Jer teoretski modeli koji uvek rade u rešenjima na kraju knjige, oni su super, ali strašno je pričati o izvorima grešaka kada se teorija ne poklapa sa praktičnim. Jer su ti razgovori bili toliko vredni, među najvrednijima.
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.
Tako da sam ovde da izvestim o zabavnim dobicima sa učenicima koju su došli prethodno instalirani sa ovim virusima, prvog dana predavanja. Ovo su deca koja sada, posle jednog polugodišta, ako stavim nešto na tablu, totalno novo, strano, oni će da pričaju o tome 3-4 minuta više nego na početku godine, što je baš zabavno. Više nemamo averziju prema tekstualnim problemima, jer smo redefinisali šta je tekstualni problem. Nismo više zastrašeni matematikom, jer polako redefinišemo šta je matematika. Ovo je bilo baš 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.
Ohrabrujem nastavnike matematike sa kojima pričam da koriste multimediju, jer to uvodi realni život u vašu učionicu sa visokom rezolucijom i bojom, da podstičemo intuiciju učenika da dođe na taj nivo, da pitate najkraća pitanja koja možete i da dopustite da se ta specifičnija pitanja jave u razgovoru, da dozvolite učenicima da sagrade problem, jer je Ajnštajn rekao tako, i na kraju, sve u svemu, da manje pomažete, jer knjiga vam pomaže na pogrešne načine. Ona vas vadi iz obaveze da strpljivo rešavate problem i matematički razmišljate, da manje pomažete.
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?
I zato je ovo neverovatno vreme da budete nastavnik matematike jer imamo alate da stvorimo program visokog kvaliteta u našem prednjem džepu. To je sveprisutno i dosta jeftino. I alat za njihovu slobodnu distribuciju sa otvorenim licencama takođe nikada nije bilo jeftinije i sveprisutnije. Postavio sam seriju klipova na mom blogu ne tako davno, i imali su 6 000 pregleda za 2 nedelje. I dalje dobijam mailove od profesora iz zemalja koje nikada nisam posetio gde kažu: "Opa, da. Imali smo dobar razgovor o tome. Oh, i usput, evo kako sam popravio tvoju stvar", što je sjajno. Postavio sam ovaj problem skoro na mom blogu. U prodavnici, u koji red da stanete, onaj koji ima jedna kolica i 19 stvari, ili onaj sa 4 kolica i 3, 5, 2 i jednom stvari. I linearno modeliranje ovde uključeno, bila je dobra stvar za moje učenike, ali me je na kraju dovelo u "Dobro jutro Ameriko" nekoliko nedelja kasnije, što je bizarno.
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)
I iz svega ovoga mogu samo da zaključim da su ljudi, ne samo učenici, željni ovoga. Matematika daje smisao svetu. Matematika je rečnik vaše intuicije. Tako da vas podržavam, kako god da učestvujete u obrazovanju, bilo da ste učenik, roditelj, nastavnik, stvarate pravila, svejedno, insistirajte na boljem matematičkom programu. Treba nam više strpljivih rešavača problema. Hvala vam.