 If Socrates Had Two Fingers         The following conversation took place between Rick Garlikov and a class of third graders. The goal was to lead the students to discovery of binary numbers through the socratic method of question-asking. (Garlikov's full experiences can be found at http://www.Garlikov.com/Soc_Meth.html.)

RG: How many is this? [I held up ten fingers.]

TEN

RG: 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

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

2 x 5 [inspired by the last idea]

RG: 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

RG: One more?

X [Roman numeral]

RG: [I point to the word "ten"]. What is this? (This step marks a change in direction of the discussion. You may want to point this out to the students.)

THE WORD TEN

RG: What are written words made up of?

LETTERS

RG: How many letters are there in the English alphabet?

26

RG: How many words can you make out of them?

ZILLIONS

RG: [Pointing to the number "10"] What is this way of writing numbers made up of?

NUMERALS

RG: How many numerals are there?

NINE / TEN

RG: Which, nine or ten?

TEN

RG: 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

RG: How many numbers can you make out of these numerals?

MEGA-ZILLIONS, INFINITE, LOTS

RG: How come we have ten numerals? Could it be because we have 10 fingers?

COULD BE

RG: What if we were aliens with only two fingers? How many numerals might we have?

2

RG: How many numbers could we write out of 2 numerals?

NOT MANY /
[one kid:] THERE WOULD BE A PROBLEM

RG: What problem?

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

RG: [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]

RG: 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...]

RG: 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

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

0 ZERO
1 ONE
[silence]

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

WRITE DOWN "ONE, ZERO"

RG: Why?

[almost in unison] I DON'T KNOW; THAT'S JUST THE WAY YOU WRITE "TEN"

RG: 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

RG: What do you call that column you put it in?

TENS

RG: Why do you call it that?

DON'T KNOW

RG: Well, what does this 1 and this 0 mean when written in these columns?

1 TEN AND NO ONES

RG: But why is this a ten? Why is this [pointing] the ten's column?

DON'T KNOW; IT JUST IS!

RG: 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

RG: Could that be why it is called the ten's column?! What is the first number that needs the next column?

100

RG: And what column is that?

HUNDREDS

RG: After you write 19, what do you have to change to write down 20?

9 to a 0 and 1 to a 2

RG: Meaning then 2 tens and no ones, right, because 2 tens are ___?

TWENTY

RG: First number that needs a fourth column?

ONE THOUSAND

RG: What column is that?

THOUSANDS

RG: Okay, let's go back to our two-fingered aliens arithmetic. We have

 0 zero 1 one

RG: 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

RG: What should we call it?

TWO'S COLUMN?

RG: Right! Because the first number we need it for is ___?

TWO

RG: So what do we put in the two's column? How many two's are there in two?

1

RG: And how many one's extra?

ZERO

RG: So then two looks like this: [pointing to "10"], right?

RIGHT, BUT THAT SURE LOOKS LIKE TEN.

RG: 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

RG: 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

RG: Ok, I'm teaching you different. What is "1, 0" here?

TWO

RG: Hard to see it that way, though, right?

RIGHT

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

THREE

RG: How do we write it with our numerals?

We need one "TWO" and a "ONE"

RG: [I write down 11 for them] So we have

 0 zero 1 one 10 two 11 three

RG: Uh oh, now we're out of numerals again. How do we get to four?

START A NEW COLUMN!

RG: Call it what?

THE FOUR'S COLUMN

RG: Call it out to me; what do I write?

ONE, ZERO, ZERO

[I write "100 four" under the other numbers]

RG: Next?

ONE, ZERO, ONE

I write "101 five"

RG: 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

RG: And we get?

ONE, ONE, ZERO

RG: 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"

RG: Which is ____?

SIX

RG: Next? Seven?

ONE, ONE, ONE

I write "111 seven"

RG: 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

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

MEGA-ZILLIONS / ALL OF THEM

RG: So who uses this stuff?

NOBODY/ ALIENS

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

NO WE DON'T

RG: Yes you do. Any ideas where?

NO

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

A SWITCH

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

TWO

RG: What could you call these positions?

ON AND OFF/ UP AND DOWN

RG: 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!