Creating the teachable moment
- Written by Andy W
- Category: Blog ideas
There is nothing more powerul than helping the student find their own answer. I call this an Insight
This is a very rewading experiment conducted by a teacher of mathmatics.
The experiment was to see whether this teacher could teach a class of students binary arithmetic (arithmetic using only two numbers, 0 and 1) ...And to teach them by only asking the class questions!
None of the students (@14yrs old) had been introduced to binary arithmetic before. Though the ostensible subject matter was binary arithmetic, my primary interest was to give a demonstration to the teacher of the power and benefit of the Socratic method where it is applicable.
Original work is published here: http://www.garlikov.com/Soc_Meth.html
The teacher introduced the lesson as follows: " what I am really here for today is to try an experiment with you. I am the subject of the experiment, not you. I want to see whether I can teach you a whole new kind of arithmetic only by asking you questions..."
"I won't be allowed to tell you anything about it, just ask you things. When you think you know an answer, just call it out. You won't need to raise your hands and wait for me to call on you; that takes too long." [This took them a while to adapt to. They kept raising their hands; though after a while they simply called out the answers while raising their hands.]
Here we go.
1) "How many is this?" [I held up ten fingers.]
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
3) Who can write ten another way? [They hesitate than some hands go up. I toss the chalk to another kid.]
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4) Another way?
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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?
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?
10) How many letters are there in the English alphabet?
11) How many words can you make out of them?
12) [Pointing to the number "10"] What is this way of writing numbers made up of?
13) How many numerals are there?
NINE / TEN
14) Which, nine or ten?
15) Starting with zero, what are they? [They call out, I write them in the following way.]
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?
18) What if we were aliens with only two fingers? How many numerals might we have?
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.
24) Okay, what can we write as we count? [I write as they call out answers.]
25) Is that it? What do we do on this planet when we run out of numerals at 9?
WRITE DOWN "ONE, ZERO"
[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?
29) Why do you call it that?
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?
33) Could that be why it is called the ten's column?! What is the first number that needs the next column?
34) And what column is that?
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 ___?
37) First number that needs a fourth column?
38) What column is that?
39) Okay, let's go back to our two-fingered aliens arithmetic. We have
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?
41) Right! Because the first number we need it for is ___?
42) So what do we put in the two's column? How many two's are there in two?
43) And how many one's extra?
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?
48) Hard to see it that way, though, right?
49) Try to get used to it; the alien children do. What number comes next?
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
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]
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 ____?
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 ____?
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:]
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?
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].
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:
x 11 times three
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?
65) And how much is two times three in real life?
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?
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?
69) So who uses this stuff?
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?
72) [I walk over to the light switch and, pointing to it, ask:] What is this?
73) [I flip it off and on a few times.] How many positions does it have?
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?