Um professor que não responde nada, ensina alguma coisa?

O texto abaixo é uma transcrição (em inglês) de uma aula especial. Uma experiência de ensino alternativo, com 22 alunos na faixa dos 10 anos de idade.

O professor era um convidado (Rick Garlikov), não conhecia as crianças e o objetivo era fazer uma aula inteira apenas com perguntas. Uma aula do avesso, onde o professor é quem pergunta e os alunos respondem e explicam. Uma hora inteira utilizando o mesmo método que Sócrates usava, a maiêutica (ou método socrático mesmo). O tema da aula foi “Código Binário”(!), a linguagem utilizada por computadores e que usa apenas os algarismos 0 e 1.

Recapitulando a encrenca na mão do Sr. Garlikov: 22 alunos, média 10 anos de idade, só podia fazer perguntas e tinha que apresentar a aritmética binária em 50 minutos.

A aposta entre os professores do colégio era que no máximo 2 ou 3 crianças seriam capazes de acompanhar o professor até o final.

A seguir você vira um dos alunos e vê como funciona isso.

1) “How many is this?” [I held up ten fingers.]

TEN

2) “Who can write that on the board?” [virtually all hands up; I toss the chalk to one kid and indicate for her to come up and do it]. She writes

10

3) Who can write ten another way? [They hesitate than some hands go up. I toss the chalk to another kid.]

4) Another way?

.

5) Another way?

.

2 x 5 [inspired by the last idea]

6) That’s very good, but there are lots of things that equal ten, right? [student nods agreement], so I’d rather not get into combinations that equal ten, but just things that represent or sort of mean ten. That will keep us from having a whole bunch of the same kind of thing. Anybody else?

TEN

7) One more?

X [Roman numeral]

8) [I point to the word “ten”]. What is this?

THE WORD TEN

9) What are written words made up of?

LETTERS

10) How many letters are there in the English alphabet?

26

11) How many words can you make out of them?

ZILLIONS

12) [Pointing to the number “10”] What is this way of writing numbers made up of?

NUMERALS

13) How many numerals are there?

NINE / TEN

14) Which, nine or ten?

TEN

15) Starting with zero, what are they? [They call out, I write them in the following way.]

0
1
2
3
4
5
6
7
8
9

16) How many numbers can you make out of these numerals?

MEGA-ZILLIONS, INFINITE, LOTS

17) How come we have ten numerals? Could it be because we have 10 fingers?

COULD BE

18) What if we were aliens with only two fingers? How many numerals might we have?

2

19) How many numbers could we write out of 2 numerals?

NOT MANY /

[one kid:] THERE WOULD BE A PROBLEM

20) What problem?

THEY COULDN’T DO THIS [he holds up seven fingers]

21) [This strikes me as a very quick, intelligent insight I did not expect so suddenly.] But how can you do fifty five?

[he flashes five fingers for an instant and then flashes them again]

22) How does someone know that is not ten? [I am not really happy with my question here but I don’t want to get side-tracked by how to logically try to sign numbers without an established convention. I like that he sees the problem and has announced it, though he did it with fingers instead of words, which complicates the issue in a way. When he ponders my question for a second with a “hmmm”, I think he sees the problem and I move on, saying…]

23) Well, let’s see what they could do. Here’s the numerals you wrote down [pointing to the column from 0 to 9] for our ten numerals. If we only have two numerals and do it like this, what numerals would we have.

0, 1

24) Okay, what can we write as we count? [I write as they call out answers.]

0             ZERO
1             ONE
[silence]

25) Is that it? What do we do on this planet when we run out of numerals at 9?

WRITE DOWN “ONE, ZERO”

26) Why?

[almost in unison] I DON’T KNOW; THAT’S JUST THE WAY YOU WRITE “TEN”

27) You have more than one numeral here and you have already used these numerals; how can you use them again?

WE PUT THE 1 IN A DIFFERENT COLUMN

28) What do you call that column you put it in?

TENS

29) Why do you call it that?

DON’T KNOW

30) Well, what does this 1 and this 0 mean when written in these columns?

1 TEN AND NO ONES

31) But why is this a ten? Why is this [pointing] the ten’s column?

DON’T KNOW; IT JUST IS!

32) I’ll bet there’s a reason. What was the first number that needed a new column for you to be able to write it?

TEN

33) Could that be why it is called the ten’s column?! What is the first number that needs the next column?

100

34) And what column is that?

HUNDREDS

35) After you write 19, what do you have to change to write down 20?

9 to a 0 and 1 to a 2

36) Meaning then 2 tens and no ones, right, because 2 tens are ___?

TWENTY

37) First number that needs a fourth column?

ONE THOUSAND

38) What column is that?

THOUSANDS

39) Okay, let’s go back to our two-fingered aliens arithmetic. We have

0          zero
1          one.

What would we do to write “two” if we did the same thing we do over here [tens] to write the next number after you run out of numerals?

START ANOTHER COLUMN

40) What should we call it?

TWO’S COLUMN?

41) Right! Because the first number we need it for is ___?

TWO

42) So what do we put in the two’s column? How many two’s are there in two?

1

43) And how many one’s extra?

ZERO

44) So then two looks like this: [pointing to “10”], right?

RIGHT, BUT THAT SURE LOOKS LIKE TEN.

45) No, only to you guys, because you were taught it wrong [grin] — to the aliens it is two. They learn it that way in pre-school just as you learn to call one, zero [pointing to “10”] “ten”. But it’s not really ten, right? It’s two — if you only had two fingers. How long does it take a little kid in pre-school to learn to read numbers, especially numbers with more than one numeral or column?

TAKES A WHILE

46) Is there anything obvious about calling “one, zero” “ten” or do you have to be taught to call it “ten” instead of “one, zero”?

HAVE TO BE TAUGHT IT

47) Ok, I’m teaching you different. What is “1, 0” here?

TWO

48) Hard to see it that way, though, right?

RIGHT

49) Try to get used to it; the alien children do. What number comes next?

THREE

50) How do we write it with our numerals?

We need one “TWO” and a “ONE”

[I write down 11 for them] So we have

0         zero
1          one
10          two
11        three

51) Uh oh, now we’re out of numerals again. How do we get to four?

START A NEW COLUMN!

52) Call it what?

THE FOUR’S COLUMN

53) Call it out to me; what do I write?

ONE, ZERO, ZERO

[I write “100       four” under the other numbers]

54) Next?

ONE, ZERO, ONE

I write “101         five”

55) Now let’s add one more to it to get six. But be careful. [I point to the 1 in the one’s column and ask] If we add 1 to 1, we can’t write “2”, we can only write zero in this column, so we need to carry ____?

ONE

56) And we get?

ONE, ONE, ZERO

57) Why is this six? What is it made of? [I point to columns, which I had been labeling at the top with the word “one”, “two”, and “four” as they had called out the names of them.]

a “FOUR” and a “TWO”

58) Which is ____?

SIX

59) Next? Seven?

ONE, ONE, ONE

I write “111 seven”

60) Out of numerals again. Eight?

NEW COLUMN; ONE, ZERO, ZERO, ZERO

I write      “1000 eight”

[We do a couple more and I continue to write them one under the other with the word next to each number, so we have:]

0          zero
1          one
10          two
11          three
100          four
101          five
110          six
111          seven
1000          eight
1001          nine
1010          ten

61) So now, how many numbers do you think you can write with a one and a zero?

MEGA-ZILLIONS ALSO/ ALL OF THEM

62) Now, let’s look at something. [Point to Roman numeral X that one kid had written on the board.] Could you easily multiply Roman numerals? Like MCXVII times LXXV?

NO

63) Let’s see what happens if we try to multiply in alien here. Let’s try two times three and you multiply just like you do in tens [in the “traditional” American style of writing out multiplication].

10          two
x 11 times   three

They call out the “one, zero” for just below the line, and “one, zero, zero” for just below that and so I write:

10          two
x 11 times   three
10
100
110

64) Ok, look on the list of numbers, up here [pointing to the “chart” where I have written down the numbers in numeral and word form] what is 110?

SIX

65) And how much is two times three in real life?

SIX

66) So alien arithmetic works just as well as your arithmetic, huh?

LOOKS LIKE IT

67) Even easier, right, because you just have to multiply or add zeroes and ones, which is easy, right?

YES!

68) There, now you know how to do it. Of course, until you get used to reading numbers this way, you need your chart, because it is hard to read something like “10011001011” in alien, right?

RIGHT

69) So who uses this stuff?

NOBODY/ ALIENS

70) No, I think you guys use this stuff every day. When do you use it?

NO WE DON’T

71) Yes you do. Any ideas where?

NO

72) [I walk over to the light switch and, pointing to it, ask:]    What is this?

A  SWITCH

73) [I flip it off and on a few times.]   How many positions does it have?

TWO

74) What could you call these positions?

ON AND OFF/ UP AND DOWN

75) If you were going to give them numbers what would you call them?

ONE AND TWO/

[one student] OH!! ZERO AND ONE!

[other kids then:] OH, YEAH!

76) You got that right. I am going to end my experiment part here and just tell you this last part.

Computers and calculators have lots of circuits through essentially on/off switches, where one way represents 0 and the other way, 1. Electricity can go through these switches really fast and flip them on or off, depending on the calculation you are doing. Then, at the end, it translates the strings of zeroes and ones back into numbers or letters, so we humans, who can’t read long strings of zeroes and ones very well can know what the answers are.

[at this point one of the kid’s in the back yelled out, OH! NEEEAT!!]

I don’t know exactly how these circuits work; so if your teacher ever gets some electronics engineer to come into talk to you, I want you to ask him what kind of circuit makes multiplication or alphabetical order, and so on. And I want you to invite me to sit in on the class with you.

Now, I have to tell you guys, I think you were leading me on about not knowing any of this stuff. You knew it all before we started, because I didn’t tell you anything about this — which by the way is called “binary arithmetic”, “bi” meaning two like in “bicycle”. I just asked you questions and you knew all the answers. You’ve studied this before, haven’t you?

NO, WE HAVEN’T. REALLY.

Then how did you do this? You must be amazing. By the way, some of you may want to try it with other sets of numerals. You might try three numerals 0, 1, and 2. Or five numerals. Or you might even try twelve 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ~, and ^ — see, you have to make up two new numerals to do twelve, because we are used to only ten. Then you can check your system by doing multiplication or addition, etc. Good luck.

After the part about John Glenn, the whole class took only 25 minutes.

Their teacher told me later that after I left the children talked about it until it was time to go home.

 

IMG BY:  gualtiero boffi/Shutterstock

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Wagner Brenner
Fundador e editor do Update or Die.

20 Comments

  1. Marcos Cavalcanti, para você que gosta de perguntas, números e dos desafios do aprendizado.

  2. incrível! via Breno Frias.

  3. iria precisar do triplo de tempo de aulas que temos hoje.

  4. Com certeza, uma vez que agindo assim ele faz com que os alunos busquem em suas memórias o que já aprenderam, ou pesquisem sobre o que ainda não conhecem.

  5. Um professor que não responde nada, ensina alguma coisa? Por incrível que pareça, sim! http://is.gd/Br0XAo

  6. muito interessante.. mas é possível com matemática apenas, certo? como vc levaria um aluno a deduzir a lei da gravidade? jogamos uma maça na cabeça de cada um? 🙂

  7. Interessante! RT @alpn00: utilizando da maiêutica, crianças de dez anos aprenderam o código binário: http://is.gd/Br0XAo (via @kenmori)

  8. utilizando da maiêutica, crianças de dez anos aprenderam o código binário: http://is.gd/Br0XAo via @kenmori

  9. Um aluno enviou-me este post dizendo que ao final tinha um sorriso nos lábios.
    Eu, um sorriso na alma. Pela dupla satisfação de ver uma lição de inteligência e de recebê-la – de um aluno.

  10. melhor post do update or die! nota 1010.

  11. Diversão e estímulo pra descobertas são grandes ” anabolizantes” pro cérebro!

  12. Fantástico! O melhor de tudo isso foi a “desconfiança” do professor de que os alunos já sabiam tudo aquilo antes que ele explicasse. Imagino a auto-estima e a satisfação que as crianças devem ter sentido. Com certeza os pais ouviram essas explicações por dias. 🙂

    Tá aí a prova de que para fazer o aluno se interessar, discutir e participar de uma aula não é necessário nenhuma ferramenta nova. É a mudança de postura da escola/professor que conta. Com certeza as crianças de 10 anos de hoje – inquietas, inteligentes e conectadas – se participassem de uma atividade dessas, ficariam animadas para assistir a próxima aula.

    Post fantástico. Obrigada por compartilhar.

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