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. That is my interest here as
well. I chose binary arithmetic as the vehicle for that because it is
something very difficult for children, or anyone, to understand when it is
taught normally; and I believe that a demonstration of a method that can teach
such a difficult subject easily to children and also capture their enthusiasm
about that subject is a very convincing demonstration of the value of the
method.
This was to be the Socratic method in what I consider its purest form,
where questions (and only questions) are used to arouse curiosity and at the
same time serve as a logical, incremental, step-wise guide that enables
students to figure out about a complex topic or issue with their own thinking
and insights.
When I got to the classroom for the binary math experiment, students were
giving reports on famous people and were dressed up like the people they were
describing. The student I came in on was reporting on John Glenn, but he had
not mentioned the dramatic and scary problem of that first American trip in
orbit. I asked whether anyone knew what really scary thing had happened on
John Glenn's flight, and whether they knew what the flight was. Many said a
trip to the moon, one thought Mars. I told them it was the first full earth
orbit in space for an American. Then someone remembered hearing about
something wrong with the heat shield, but didn't remember what. By now they
were listening intently. I explained about how a light had come on that
indicated the heat shield was loose or defective and that if so, Glenn would
be incinerated coming back to earth. But he could not stay up there alive
forever and they had nothing to send up to get him with. The engineers finally
determined, or hoped, the problem was not with the heat shield, but with the
warning light. They thought it was what was defective. Glenn came down. The
shield was ok; it had been just the light. They thought that was neat.
"But 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.]
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 then 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.
(Rick
Garlikov's) Views About This Whole
Episode
Students do not get bored or lose concentration if they are actively
participating. Almost all of these children participated the whole time; often
calling out in unison or one after another. If necessary, I could have asked if
anyone thought some answer might be wrong, or if anyone agreed with a particular
answer. You get extra mileage out of a given question that way. I did not have to
do that here. Their answers were almost all immediate and very good. If necessary,
you can also call on particular students; if they don't know, other students will
bail them out. Calling on someone in a non-threatening way tends to activate
others who might otherwise remain silent. That was not a problem with these kids.
Remember, this was not a "gifted" class. It was a normal suburban third grade of
whom two teachers had said only a few students would be able to understand the
ideas.
The topic was "twos", but I think they learned just as much about the "tens"
they had been using and not really understanding.
This method takes a lot of energy and concentration when you are doing it fast,
the way I like to do it when beginning a new topic. A teacher cannot do this for
every topic or all day long, at least not the first time one teaches particular
topics this way. It takes a lot of preparation, and a lot of thought. When it goes
well, as this did, it is so exciting for both the students and the teacher that it
is difficult to stay at that peak and pace or to change gears or topics. When it
does not go as well, it is very taxing trying to figure out what you need to
modify or what you need to say. I practiced this particular sequence of
questioning a little bit one time with a first grade teacher. I found a flaw in my
sequence of questions. I had to figure out how to correct that. I had time to
prepare this particular lesson; I am not a teacher but a volunteer; and I am not a
mathematician. I came to the school just to do this topic that one period.
I did this fast. I personally like to do new topics fast originally and then
re-visit them periodically at a more leisurely pace as you get to other ideas or
circumstances that apply to, or make use of, them. As you re-visit, you fine tune.
The chief benefits of this method are that it excites students' curiosity and
arouses their thinking, rather than stifling it. It also makes teaching more
interesting, because most of the time, you learn more from the students -- or by
what they make you think of -- than what you knew going into the class. Each group
of students is just enough different, that it makes it stimulating. It is a very
efficient teaching method, because the first time through tends to cover the topic
very thoroughly, in terms of their understanding it. It is more efficient for
their learning then lecturing to them is, though, of course, a teacher can lecture
in less time.
It gives constant feed-back and thus allows monitoring of the students'
understanding as you go. So you know what problems and misunderstandings or lack
of understandings you need to address as you are presenting the material. You do
not need to wait to give a quiz or exam; the whole thing is one big quiz as you
go, though a quiz whose point is teaching, not grading. Though, to repeat, this is
teaching by stimulating students' thinking in certain focused areas, in order to
draw ideas out of them; it is not "teaching" by pushing ideas into students that
they may or may not be able to absorb or assimilate. Further, by quizzing and
monitoring their understanding as you go along, you have the time and opportunity
to correct misunderstandings or someone's being lost at the immediate time, not at
the end of six weeks when it is usually too late to try to "go back" over the
material. And in some cases their ideas will jump ahead to new material so that
you can meaningfully talk about some of it "out of (your!) order" (but in an order
relevant to them). Or you can tell them you will get to exactly that in a little
while, and will answer their question then. Or suggest they might want to think
about it between now and then to see whether they can figure it out for themselves
first. There are all kinds of options, but at least you know the material is
"live" for them, which it is not always when you are lecturing or just telling
them things or they are passively and dutifully reading or doing worksheets or
listening without thinking.
If you can get the right questions in the right sequence, kids in the whole
intellectual spectrum in a normal class can go at about the same pace without
being bored; and they can "feed off" each others' answers. Gifted kids may have
additional insights they may or may not share at the time, but will tend to
reflect on later. This brings up the issue of teacher expectations. From what I
have read about the supposed sin of tracking, one of the main complaints is that
the students who are not in the "top" group have lower expectations of themselves
and they get teachers who expect little of them, and who teach them in boring ways
because of it. So tracking becomes a self-fulfilling prophecy about a kid's
educability; it becomes dooming. That is a problem, not with tracking as such, but
with teacher expectations of students (and their ability to teach). These kids
were not tracked, and yet they would never have been exposed to anything like this
by most of the teachers in that school, because most felt the way the two did
whose expectations I reported. Most felt the kids would not be capable enough and
certainly not in the afternoon, on a Friday near the end of the school year yet.
One of the problems with not tracking is that many teachers have almost as low
expectations of, and plans for, students grouped heterogeneously as they do with
non-high-end tracked students. The point is to try to stimulate and challenge all
students as much as possible. The Socratic method is an excellent way to do that.
It works for any topics or any parts of topics that have any logical natures at
all. It does not work for unrelated facts or for explaining conventions, such as
the sounds of letters or the capitals of states whose capitals are more the result
of historical accident than logical selection.
Of course, you will notice these questions are very specific, and as logically
leading as possible. That is part of the point of the method. Not just any
question will do, particularly not broad, very open ended questions, like "What is
arithmetic?" or "How would you design an arithmetic with only two numbers?" (or if
you are trying to teach them about why tall trees do not fall over when the wind
blows "what is a tree?"). Students have nothing in particular to focus on when you
ask such questions, and few come up with any sort of interesting answer.
And it forces the teacher to think about the logic of a topic, and how to make
it most easily assimilated. In tandem with that, the teacher has to try to
understand at what level the students are, and what prior knowledge they may have
that will help them assimilate what the teacher wants them to learn. It emphasizes
student understanding, rather than teacher presentation; student intake,
interpretation, and "construction", rather than teacher output. And the point of
education is that the students are helped most efficiently to learn by a teacher,
not that a teacher make the finest apparent presentation, regardless of what
students might be learning, or not learning. I was fortunate in this class that
students already understood the difference between numbers and numerals, or I
would have had to teach that by questions also. And it was an added help that they
had already learned Roman numerals. It was also most fortunate that these students
did not take very many, if any, wrong turns or have any firmly entrenched
erroneous ideas that would have taken much effort to show to be mistaken.
I took a shortcut in question 15 although I did not have to; but I did it
because I thought their answers to questions 13 and 14 showed an understanding
that "0" was a numeral, and I didn't want to spend time in this particular lesson
trying to get them to see where "0" best fit with regard to order. If they had
said there were only nine numerals and said they were 1-9, then you could ask how
they could write ten numerically using only those nine, and they would quickly
come to see they needed to add "0" to their list of numerals.
These are the four critical points about the questions: 1) they must be
interesting or intriguing to the students; they must lead by 2) incremental and 3)
logical steps (from the students' prior knowledge or understanding) in order to be
readily answered and, at some point, seen to be evidence toward a conclusion, not
just individual, isolated points; and 4) they must be designed to get the student
to see particular points. You are essentially trying to get students to use their
own logic and therefore see, by their own reflections on your questions, either
the good new ideas or the obviously erroneous ideas that are the consequences of
their established ideas, knowledge, or beliefs. Therefore you have to know or to
be able to find out what the students' ideas and beliefs are. You cannot ask just
any question or start just anywhere.
It is crucial to understand the difference between "logically" leading
questions and "psychologically" leading questions. Logically leading questions
require understanding of the concepts and principles involved in order to be
answered correctly; psychologically leading questions can be answered by students'
keying in on clues other than the logic of the content. Question 39 above is
psychologically leading, since I did not want to cover in this
lesson the
concept of value-representation but just wanted to use "columnar-place" value, so
I psychologically led them into saying "Start another column" rather than getting
them to see the reasoning behind columnar-place as merely one form of value
representation. I wanted them to see how to use columnar-place value logically
without trying here to get them to totally understand
its logic. (A
common form of value-representation that is not "place" value is color value in
poker chips, where colors determine the value of the individual chips in ways
similar to how columnar place does it in writing. For example if white chips are
worth "one" unit and blue chips are worth "ten" units, 4 blue chips and 3 white
chips is the same value as a "4" written in the "tens" column and a "3" written in
the "ones" column for almost the same reasons.)
For the Socratic method to work as a teaching tool and not just as a magic trick
to get kids to give right answers with no real understanding, it is crucial that
the important questions in the sequence must be logically leading rather than
psychologically leading. There is no magic formula for doing this, but one of the
tests for determining whether you have likely done it is to try to see whether
leaving out some key steps still allows people to give correct answers to things
they are not likely to really understand. Further, in the case of binary numbers,
I found that when you used this sequence of questions with impatient or
math-phobic adults who didn't want to have to think but just wanted you to "get to
the point", they could not correctly answer very far into even the above sequence.
That leads me to believe that answering most of these questions correctly,
requires understanding of the topic rather than picking up some "external" sorts
of clues in order to just guess correctly. Plus, generally when one uses the
Socratic method, it tends to become pretty clear when people get lost and are
either mistaken or just guessing. Their demeanor tends to change when they are
guessing, and they answer with a questioning tone in their voice. Further, when
they are logically understanding as they go, they tend to say out loud insights
they have or reasons they have for their answers. When they are just guessing,
they tend to just give short answers with almost no comment or enthusiasm. They
don't tend to want to sustain the activity.
Finally, two of the interesting, perhaps side, benefits of using the Socratic
method are that it gives the students a chance to experience the attendant joy and
excitement of discovering (often complex) ideas on their own. And it gives
teachers a chance to learn how much more inventive and bright a great many more
students are than usually appear to be when they are primarily passive.
Used with permission: C.F.Garlikov
http://www.garlikov.com/Soc_Meth.html
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