Irrational Math Learn Direct Math %D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%20%D1%81%D0%B5%D1%80%D0%BF%D0%B8%D0%BD%D1%81%D0%BA%D0%BE%D0%B3%D0%BE%20java
3rd Grade Math Grade 6 Math Worksheets Mathlearndirect P Learn Ko Home Article518 Math Learn Direct Www Mathlearndirect R K K Calculus O O Gradient Math Szh 1 Math Learn Direct The Socratic Method
3rd Grade Math Grade 6 Math Worksheets Mathlearndirect P Learn Ko Home Article518 Math Learn Direct
Www Mathlearndirect R K K Calculus O O Gradient Math Szh 1 Math Learn Direct The Socratic Method
Www Mathlearndirect R K K Calculus O O Gradient Math Szh 1 Math Learn Direct
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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.
. . . . . . . . . . . . . .
My 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.
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