Other Foundations of Math secondary teachers?
math...@mikeskettle.com
Hello,
I am a new teacher. I am teaching a class called Foundations of Math
for 9th graders who are not ready for algebra. I have a small class
size of 17 students. All of my students are ELL and most are having a
hard time with even the basic concepts of math.
My big problem is that the book is terrible and there is no curriculum
outline for this class. The nice thing is that I have the freedom to
do almost anything.
I will have an entire year with these students and we are on a block
schedule so we spend 95 minutes a day together. I would love to talk
to other teachers that are in similar situations. What do you do with
students for this amount of time when there is no preset curriculum?
What sequence do you take the kids through when they aren't strong on
the basics? How often do you do formal assessments? Do you
differentiate instruction or do you provide traditional direct
instruction?
Even though my students are ELL, their English is not bad. Also, they
all seem to really want to learn. When I provide them with a
challenging project, they work quietly and vigorously.
Again, it would be nice if there were other teachers in this situation.
If you are, please say hello.
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Brian Harvey
math...@mikeskettle.com writes:
>I am a new teacher. I am teaching a class called Foundations of Math
>students for this amount of time when there is no preset curriculum?
Consider teaching them algebra.
Elementary school math (arithmetic) depends mainly on two skills:
(1) memorizing arbitrary stuff; (2) tolerating the arbitrariness of the
arbitrary stuff. [Yes, I know that once you understand a lot of math,
both the number facts and the multi-digit algorithms become non-arbitrary.
But there isn't one kid in 20 who understands all that.]
Algebra is completely different. There are *reasons* for things. The main
skill is logical reasoning. If you think your kids don't have that, watch
them playing computer games.
Teaching them arithmetic one more time (even if disguised as checkbook
balancing, or whatever the latest "real application" fad is) will just
give them one more chance to fail.
The trick is to convince *them* that this isn't going to be just the same
stuff for them to fail at again. Maybe start with something that doesn't
have any numbers at all, such as logic puzzles. (Leave out the ones about
relative ages. :-)
The other possibility is to teach them computer programming. They
exercise the same skills, but see an immediate result of their work.
Of course, for this you need computers -- do you have them available?
J. Frost
In article <dhqe43$4sl$1...@abbenay.CS.Berkeley.EDU>,
Brian Harvey <b...@cs.berkeley.edu> wrote:
>
>math...@mikeskettle.com writes:
>>I am a new teacher. I am teaching a class called Foundations of Math
>>for 9th graders who are not ready for algebra. [...] What do you do with
>>students for this amount of time when there is no preset curriculum?
>
>Consider teaching them algebra.
>
>Elementary school math (arithmetic) depends mainly on two skills:
>(1) memorizing arbitrary stuff; (2) tolerating the arbitrariness of the
>arbitrary stuff. [Yes, I know that once you understand a lot of math,
>both the number facts and the multi-digit algorithms become non-arbitrary.
>But there isn't one kid in 20 who understands all that.]
>
>Algebra is completely different. There are *reasons* for things. The main
>skill is logical reasoning. If you think your kids don't have that, watch
>them playing computer games.
>
>Teaching them arithmetic one more time (even if disguised as checkbook
>balancing, or whatever the latest "real application" fad is) will just
>give them one more chance to fail.
>
>The trick is to convince *them* that this isn't going to be just the same
>stuff for them to fail at again. Maybe start with something that doesn't
>have any numbers at all, such as logic puzzles. (Leave out the ones about
>relative ages. :-)
>
>The other possibility is to teach them computer programming. They
>exercise the same skills, but see an immediate result of their work.
>Of course, for this you need computers -- do you have them available?
My wife has taught that sort of class at the high school level several times,
and has found that Harold Jacobs' book "Mathematics, a Human Endeavor" is
a good place to start. The book is aimed at undergraduates who don't think
they like math, and it is a sampler of those things that don't get covered
in remedial arithmetic courses. The topics are the interesting things in
mathematics, so it can be a help in motivating an interest in starting to
learn the level of mathematics after arithmetic.
Another year of arithmetic slower and louder is likely to be a waste of
everyone's time.
Duane Bozarth
Brian Harvey wrote:
>
> math...@mikeskettle.com writes:
> >I am a new teacher. I am teaching a class called Foundations of Math
> >for 9th graders who are not ready for algebra. [...] What do you do with
> >students for this amount of time when there is no preset curriculum?
>
> Consider teaching them algebra.
>
> Elementary school math (arithmetic) depends mainly on two skills:
> (1) memorizing arbitrary stuff; (2) tolerating the arbitrariness of the
> arbitrary stuff. [Yes, I know that once you understand a lot of math,
> both the number facts and the multi-digit algorithms become non-arbitrary.
> But there isn't one kid in 20 who understands all that.]
>
> Algebra is completely different. There are *reasons* for things. The main
> skill is logical reasoning. If you think your kids don't have that, watch
> them playing computer games.
>
> Teaching them arithmetic one more time (even if disguised as checkbook
> balancing, or whatever the latest "real application" fad is) will just
> give them one more chance to fail.
But if they don't know arithmetic yet, then they'll continue to fail at
everything else...
Brian Harvey
dp_bo...@swko.dot.net writes:
>But if they don't know arithmetic yet, then they'll continue to fail at
>everything else...
No, I don't believe this is true.
It depends on what "don't know arithmetic" means. For the most part, it
means "haven't memorized the times table" and/or "get confused about the
partial products when multiplying multi-digit numbers."
Neither of those skills is the least bit necessary. Just hand them a
calculator. Giving out calculators is a lot cheaper than keeping them in
prison for most of their lives when they can't get a job because they fail
the state exit exam (20% of California seniors, according to today's paper).
And nothing in real math (by which I mean math that's about reasoning
rather than memorizing) depends on the ability to do arithmetic.
It's true that if a kid doesn't know what adding or multiplying *means*
then s/he's in trouble. So I think we should just hand out the calculators
in kindergarten, and focus the math curriculum on how to get from a word
problem to knowing which calculator button to push.
(Yes, sure, it's even better if the kid can reason *and* memorize. But some
kids just can't -- they are bright kids with a specific learning disability
about short-term memory, and we needlessly make their school lives miserable
by providing a curriculum that's entirely about memorization.)
Guess who
>Consider teaching them algebra.
>
>Elementary school math (arithmetic) depends mainly on two skills:
>(1) memorizing arbitrary stuff; (2) tolerating the arbitrariness of the
>arbitrary stuff. [Yes, I know that once you understand a lot of math,
>both the number facts and the multi-digit algorithms become non-arbitrary.
>But there isn't one kid in 20 who understands all that.]
You are right. I've asked adults, even some teachers, to divide one
fraction by another. They get it right. They invert and multiply
with no errors. Then I ask them *why* they did it that way. However,
it should not be surprising. If you learn to play the guitar or any
other musical instrument, you are first taught how to place your hands
and how to move them. You will practice scales on the piano, not
knowing why at the time, but that practice lays a firm foundation even
though the understanding is missing initially. It is the same with
all study, and is to be expected. There are always exceptions, of
course, but genius is rare.
>Algebra is completely different. There are *reasons* for things. The main
>skill is logical reasoning. If you think your kids don't have that, watch
>them playing computer games.
Not entirely completely different. Also, "logical reasoning" is also
applicable to any other study. The main present skill is in learning
the skills necessary for later realisation of the connections between
ideas. Those connections can not be made immediately, since they
might [and do] cover a lot of ground, and create a maze too deep for
the beginner. Algebra generalises the rules of arithmetic, ["x"
stands for anything"], but the rules are the same. What algebra does
do, in the beginning, is to allow the person to see more clearly what
is happening through observing [the key part of the process] how
things change and move around or stay the same. So the observer, the
"student", can see that a system not only works every time, but all of
the time, even for problems not yet done. Then, of course, it becomes
a study in itself, leaving arithmetic in the background, but still
being a basis for that understanding.
The key is still observation by the individual. There are always
those who see only x's and y's all over a page, with no apparent
connection.
Duane Bozarth
Brian Harvey wrote:
>
> dp_bo...@swko.dot.net writes:
> >But if they don't know arithmetic yet, then they'll continue to fail at
> >everything else...
>
> No, I don't believe this is true.
>
> It depends on what "don't know arithmetic" means. For the most part, it
> means "haven't memorized the times table" and/or "get confused about the
> partial products when multiplying multi-digit numbers."
>
I emphatically disagree that not knowing at least fundamental arithmetic
is essential...
mw
Thank you all for your opinions. I really appreciate the discussion of
the situation. Brian, your thoughts, although unconventional, are very
thought provoking. Some of the greatest discoveries of mankind were
made inadvertantly, not through long and tedious phases of learning and
planning. And, I don't have an all or nothing attitude. I can throw in
the basics and spice things up with algebra as often as I please. I'll
give it a try and let you know how things work.
mw
Brian Harvey
Guess who <notreal...@here.com> writes:
> If you learn to play the guitar or any
>other musical instrument, you are first taught how to place your hands
>and how to move them. You will practice scales on the piano, not
>knowing why at the time, but that practice lays a firm foundation even
>though the understanding is missing initially.
It's been a long time, but I'm pretty sure that they wanted me to understand
key signatures and circle-of-fifths from the beginning, before I had much
"firm foundation" of playing skill. (Not to mention that music teaching also
has its radical critics, for some of the same reasons as math teaching -- it
turns off more people than it turns on.)
I think, too, that the original context of this thread has been lost among
the big ideas. We are talking about a population of kids who have already
failed at learning arithmetic. So we *know for sure* that more of the same
is *not* going to do *these* kids any good. Maybe giving them some actual
mathematics won't work either, for many of them, but maybe it will, and it
certainly can't do any worse than yet another year of remedial arithmetic.
Duane Bozarth
Brian Harvey wrote:
>
> Guess who <notreal...@here.com> writes:
> > If you learn to play the guitar or any
> >other musical instrument, you are first taught how to place your hands
> >and how to move them. You will practice scales on the piano, not
> >knowing why at the time, but that practice lays a firm foundation even
> >though the understanding is missing initially.
>
> It's been a long time, but I'm pretty sure that they wanted me to understand
> key signatures and circle-of-fifths from the beginning, before I had much
> "firm foundation" of playing skill. (Not to mention that music teaching also
> has its radical critics, for some of the same reasons as math teaching -- it
> turns off more people than it turns on.)
>
> I think, too, that the original context of this thread has been lost among
> the big ideas. We are talking about a population of kids who have already
> failed at learning arithmetic. So we *know for sure* that more of the same
> is *not* going to do *these* kids any good. Maybe giving them some actual
> mathematics won't work either, for many of them, but maybe it will, and it
> certainly can't do any worse than yet another year of remedial arithmetic.
>
I guess that it also partly depends on what the class is---is it the
"cutups" or is it a real "mentally-challenged" group or something else
or all of the above?
I was thinking of apparently educatable kids who were either behind for
language or other reasons. While I don't have a problem w/ the idea of
trying some more advanced concepts, I've seen too many pushed through
that still can't do remedial arithmetic to think it's a good thing to
simply "let it slide" as unimportant.
Guess who
>I think, too, that the original context of this thread has been lost among
>the big ideas. We are talking about a population of kids who have already
>failed at learning arithmetic. So we *know for sure* that more of the same
>is *not* going to do *these* kids any good. Maybe giving them some actual
>mathematics won't work either, for many of them, but maybe it will, and it
>certainly can't do any worse than yet another year of remedial arithmetic.
You are right again. I taught every grade, every level, and really
do understand the needs of kids who have great difficulties, or at
least have a good deal of experience dealing with them. They know who
they are better than we do. Most need hands-on, and math when they
need it applied to what they are doing immediately. However, I'd not
easily accept algebra as a viable option to more, and hopefully more
appropriate application. It's simply too abstract, and kids having
difficulty with numbers that they can see will have more difficulty
with algebra that they will never use in several lifetimes. More
exciting for the teacher, perhaps, but murder for them. Even kids
with moderate difficulty have more difficulty with algebra than
arithmetic. For all of the fact that *we* can see the connection,
they by and large can not.
Brian Harvey
Guess who <notreal...@here.com> writes:
> However, I'd not
>easily accept algebra as a viable option to more, and hopefully more
>appropriate application. It's simply too abstract, and kids having
>difficulty with numbers that they can see will have more difficulty
>with algebra that they will never use in several lifetimes. More
>exciting for the teacher, perhaps, but murder for them. Even kids
>with moderate difficulty have more difficulty with algebra than
>arithmetic. For all of the fact that *we* can see the connection,
>they by and large can not.
Well, my own kid may be a counterexample. I adopted Heath at age 12;
by then he'd had a long history of school failure. Part of the reason
is that the one learning disability all the psychologists agree he has
is in short-term memory, which they say is what you need to learn the
number facts. So he came to me convinced he hates math. The schools
here put him through more and more arithmetic -- although some of it was
arithmetic with fractions, which they considered an appropriate advance
for a kid who can't multiply integers -- including, most shamefully, a
class they *called* "algebra" in which he did more remedial arithmetic.
Finally, just last year, he got his first glimpse of actual algebra, and
came home to tell me, with great surprise, that he actually enjoyed it,
"because it's logical." Meaning that you can actually figure out the
answer instead of having to remember it.
Sadly, he's now 18, impatient to get on with his life, and unlikely to
spend the time to develop any real love for the subject. Still, he
encourages me in my view that you don't need arithmetic skill to do
algebra.
I'm not saying that *every* kid who can't do arithmetic will be rescued
by algebra. Most, alas, will just go through life hating math. But
a few will be rescued, and nobody will be hurt, by the exposure to actual
mathematics.