|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
RG: How many is this? [I held up ten fingers.]
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
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?
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?
RG: One more?
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.)
RG: What are written words made up of?
RG: How many letters are there in the English alphabet?
RG: How many words can you make out of them?
RG: [Pointing to the number "10"] What is this way of writing numbers made up of?
RG: How many numerals are there?
RG: Which, nine or ten?
RG: Starting with zero, what are they? [They call out, I write them in the following way.]
RG: How many numbers can you make out of these numerals?
RG: How come we have ten numerals? Could it be because we have 10 fingers?
RG: What if we were aliens with only two fingers? How many numerals might we have?
RG: How many numbers could we write out of 2 numerals?
RG: What problem?
RG: [This strikes me as a very quick, intelligent insight I did not expect so suddenly.] But how can you do fifty five?
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.
RG: Okay, what can we write as we count? [I write as they call out answers.]
RG: Is that it? What do we do on this planet when we run out of numerals at 9?
RG: You have more than one numeral here and you have already used these numerals; how can you use them again?
RG: What do you call that column you put it in?
RG: Why do you call it that?
RG: Well, what does this 1 and this 0 mean when written in these columns?
RG: But why is this a ten? Why is this [pointing] the ten's column?
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?
RG: Could that be why it is called the ten's column?! What is the first number that needs the next column?
RG: And what column is that?
RG: After you write 19, what do you have to change to write down 20?
RG: Meaning then 2 tens and no ones, right, because 2 tens are ___?
RG: First number that needs a fourth column?
RG: What column is that?
RG: Okay, let's go back to our two-fingered aliens arithmetic. We have
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?
RG: What should we call it?
RG: Right! Because the first number we need it for is ___?
RG: So what do we put in the two's column? How many two's are there in two?
RG: And how many one's extra?
RG: So then two looks like this: [pointing to "10"], right?
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?
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"?
RG: Ok, I'm teaching you different. What is "1, 0" here?
RG: Hard to see it that way, though, right?
RG: Try to get used to it; the alien children do. What number comes next?
RG: How do we write it with our numerals?
RG: [I write down 11 for them] So we have
RG: Uh oh, now we're out of numerals again. How do we get to four?
RG: Call it what?
RG: Call it out to me; what do I write?
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 ____?
RG: And we get?
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.]
RG: Which is ____?
RG: Next? Seven?
RG: Out of numerals again. Eight?
RG: So now, how many numbers do you think you can write with a one and a zero?
RG: So who uses this stuff?
RG: No, I think you guys use this stuff every day. When do you use it?
RG: Yes you do. Any ideas where?
RG: [I walk over to the light switch and, pointing to it, ask:] What is this?
RG: [I flip it off and on a few times.] How many positions does it have?
RG: What could you call these positions?
RG: If you were going to give them numbers what would you call them?