You got it! Yup, it was a trick question! It took the class about 40 minutes before the first student got it! But the class didn't have any trouble comparing 1/8th and 50%.
Are you disagreeing with my assessment or just riffing on how much better your class was doing?
I wouldn't want to speak real confidently about things that happened 20 years ago, especially my 10 year old assessments of the mental states of other 10 year olds.
> The point of the lesson/activity is to evaluate whether the students understand that 1/8, 0.35 and 50% are comparable things.
It appeared that the teacher, in the 7th grade, was trying to teach probability, and your point about "1/8, 0.35 and 50%" is not
not significantly about probability. Further your point about "1/8, 0.35 and 50%" is too elementary for the 7th grade. Uh, at one time I was in the 6th grade, and the 7th, and even, could one believe, the 8th and do remember some of it!
My 7th grade teacher was quite comfortable giving that little puzzle problem and letting the class spend the whole period on it. It was a good little lesson.
It appears from this thread and some other sources, e.g., from
North Carolina, that probability is too difficult
for most of the K-12 faculty. So, I gave some easy lessons
on how to teach elementary probability.
I've been a student, am well qualified in probability, and
have done a lot of teaching. I know fairly well what the heck
I'm talking about.
Sorry, but that 7th grade teacher was not doing well.
Also the remarks above in this thread -- I tried to be nice
and not mention them -- on "continuous probabilities"
and "fuzzy" need to be flushed: Some probability distributions are continuous, but the 'probabilities' themselves are not. That is, we know in very fine detail just what 'continuous' means and what a 'probability' is, and 'continuous' does not apply to a 'probability'.
Further,
'fuzzy' is from a quite different subject. Saying 'fuzzy' seems
to imply that maybe some probabilities do not really exist;
no, this is a fundamental error, and quite serious. Instead,
with meager assumptions we can assume that probabilities
always exist; the issue, then, can be finding numerical
values for the probabilities, and for that we usually have
to make an 'estimate' and that is mostly in the field
of statistics and usually makes use of at least the
weak law of large numbers. That we have to make an
estimate does not justify 'fuzzy'.
From the OP, I can believe that the 7th grade students
are up to understanding probability but that the OP
is not and is not up to teaching it. Sorry 'bout that.
It appears that the K-12 teachers have been in an echo
chamber far too far from the solid material in US
universities and now are frequently circling their
wagons to defend themselves from attacks on their,
let me pick the right word, incompetence. Instead, the
teachers should learn some probability. I suggest
texts by M. Loeve (long at Berkeley), J. Neveu (from Paris and a Loeve student), K. Chung (long at Stanford), L. Breiman (a Loeve student and long at Berkeley),
or E. Cinlar (long at Princeton)? To read these texts,
start with, say, W. Rudin, 'Principles of Mathematical
Analysis' (at least once used in Harvard's Math 55), the first half of W. Rudin, 'Real and Complex
Analysis', and H. Royden, 'Real Analysis'. Royden was long at Stanford. To read those
books, be a college math major in a good program.
Then we can talk more about probability and how to
teach it. In the meanwhile, if I had a student in
that 7th grade class, I'd insist to the principal that
the class receive some competent instruction in
probability from a different teacher.
Probability is a nice subject, and it is possible to
do a good job with an elementary introduction, but
the OP showed that the content of the teaching
was a mess. E.g., there was "percent probabilities" and "theoretical probability",
and the informed mind boggles. With such teaching,
probability will be several times more difficult than
necessary; with such teaching, the students
are done real harm and later will have to "unlearn what
you have learned". So, the teaching is not just babysitting
but harmful babysitting.
You were clear enough before. You still hand wave aside the question about whether the majority of the students are ready for the material (this is separate from the question of whether they should be).
(I agree that the comparison is not probability, but understanding the comparison is what makes probability useful in everyday life...)
You are are hand waving that 7th grade students
are not yet ready for 5th and 6th grade work.
Why the heck not?
If we take your hand waving seriously, then
there are problems much more difficult and serious
than elementary probability.
I have no patience with your assumption: If
I had a child in such a class, they'd be home
with home schooling or some such alternative before the teacher first
touched chalk to the board. No patience at all.
Can't build a strong house on a rotten foundation.
The school I went to expected the students to keep
up with the right material. I have no patience with
lowering the standards to meet poor students.
> understanding the comparison is what makes probability useful in everyday life
Where can one get the really strong funny stuff someone
needs to be smoking to assume that such a triviality is "useful"? That "comparison" is just triviality
from trying to bend education down to students
not ready for 7th grade work. You are welcome to
pursue such things if you want, but any of my
children would be out'a there and home schooled,
charter schooled, privately schooled, etc. in a
nanosecond. The teacher, the principal, and the
school board would all hear from me loud, clear
messages.
There are some good reasons for a local school board:
Parents can scream bloody murder at incompetence and
trivialities. E.g., my father in law was on his
local school board, and my wife was Valedictorian,
PBK, Woodrow Wilson, 'Summa Cum Laude', two years
of NSF fellowship in one award, and Ph.D. There wasn't
a lot of patience with grotesque incompetence.
My father got transferred to a new city and for a house
first he looked at the schools. He picked what was
by a wide margin the best school in the city and then
picked a house in the district of that school. It was
a decently good school although could have been
much better.
We need much less patience with incompetence. "No
child left behind" can't be permitted to mean
leaving all the children behind.
