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Re: [math-learn] Re: Transforming formulas
Posted:
Feb 2, 2007 7:31 AM
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I think that some problems students have dealing with abstract
thinking, some of which is shown in the post below, can be dealt with
successfully to some meaningful degree. It has to do with how we say
things in our natural language.
Mathematics is very much based on the concept of the conditional p ->
q. But how we say the conditional in our natural language has
everything to do with how we psychologically process it. To help the
student understand, there is no problem, in natural language, to go
ahead and use counterfactual language to convey the conditional. Let's
call this counterfactual grammar or tense if we wish.
Example:
I could say to the students, "If you are going at 3 mph, then what is
the distance?"
Or you could use counterfactual grammar/tense/language, and say, "If
you were going at 3 mph, then what would be the distance?"
In my experience, I find that my using counterfactual language when I
explain things essentially changes everything in terms of what
abstractions students can grasp.
Note: I have found it most helpful to have the students understand
conditionals p -> q, no matter how we say them in natural language, to
just be manners of speaking that convey the idea that we will never
have the first condition p be the case without also having the second
condition q be the case. I've found this helpful in having them grasp
how, if p and q are both false, the conditional itself is still true.
(To show how counterfactual language can help, I repeat what I just
said in counterfactual mode: I've found this helpful in having them
grasp how, if p and q were both false, the conditional itself would
still be true.)
By the way, one wonders whether Piaget or his followers had thought to
consider that by just changing the language when we explain things, we
might see a difference as to what the students grasp. But regardless, I
have found through experience that using counterfactual language
changes things in a very big way as to how students respond, as to what
they grasp.
And to those who know about counterfactual conditionals in logic and
philosophy, to anticipate some things from those who think that what
I'm suggesting might be "not allowed": Even if we break some
*arbitrary* axioms put forth by some philosopher when we do this, then
I say "So what?" (Note: A given axiom is a thing we need not accept as
"handed down from on high". But, as to what truly deductively flows
from it, yes we should. I just choose to not be bound by someone's
arbitrary axiom system. I refuse to say I'm not allowed to use natural
language in a very helpful way just because some philosopher in some
ivory tower decrees through his axiom system that I'm not allowed.)
Consider the general Google search results for "counterfactuals
mathematics" (not including the quotation marks)
http://www.google.com/search?hl=en&q=counterfactuals+mathematics+
and any other Google keyword search you can come up with involving
these two words. (It's not all set in stone as to what we are allowed
to say, regardless of what some might want to tell us from the world of
philosophy. They're constantly arguing with each other, never resolving
things. We most certainly should not allow ourselves to be enslaved by
any of their decrees. Confession: Yes, I used to be really into
philosophy. But when I found it to be essentially nothing but a bunch
of never-ending argumentation that never really resolves anything,
never gives us some real facts to work with and from, I gave up on it.)
Paul
--- elkashish <forums@learningbyyourself.com> wrote:
>
> >
> > I suspect that it is possible to teach young children basic
> > manipulations without getting any understanding of what variables
> are
> > about. That is certainly happening in older students.
> >
>
> Despite what I said before about those advanced lessons, I think that
> abstract thinking is difficult for many people. Here is a made up
> conversation one might hear between a teacher and a student regarding
> variables. It is not made up to reflect how I teach it or how it
> should be taught, but the difficulty of many students have in
> grasping
> what a variable is. I am also laying a bit of a trap in thinking here
> by not saying the crucial element that can help solve the issue since
> this element is never mentioned in books <smile> [and often not
> taught
> in classes either].
>
>
> Teacher: Last time we learned that a parameter is a quantity that is
> fixed, but we don't know it's value, and therefore we write a letter
> (say a) and not a number. That parameter can get any value of course,
> depending on the conditions.
>
> Student: yeah, like when we run 3mph than our speed is the parameter,
> which if we multiply by the time we run will give us the distance,
> right? [he is repeating last class].
>
> Teacher: exactly Dan. Now, do you know how much time you ran?
>
> Dan: well, you didn't tell me.
>
> Teacher: But I told you that speed*time = distance and you figured
> it's a good formula, right?
>
> Dan: yup. But I still don't know how much time I ran.
>
> Teacher: Right. Now, how much was the distance you ran?
>
> Dan: how can I know? I don't know how much time I ran.
>
> Teacher: so let's call the time you ran 't'.
>
> Dan: Ok. But what does it mean?
>
> Teacher: it means you don't know how much time you ran, and you call
> it 't' to show that you don't know and it can be any value. We call
> it
> a variable.
>
> Student: Didn't you say we call it a parameter?
>
> Teacher: Not really. A parameter has a specific value and we only use
> a letter because we don't know what is that value. A variable can be
> any value.
>
> Student: Hmm...I'm confused.
>
> Teacher: let me show you with the example. You run 3mph, right?
>
> Student: Yup.
>
> Teacher: so if you knew how much time you ran, would you be able to
> find the distance you ran?
>
> Student:Oh, that's easy. I'd multiply 3 by the time.
>
> Teacher: great. So let's call the time 't' and multiply. What do you
> get?
>
> Student: 3t.
>
> Teacher: great, and that is equal what?
>
> Dan: the distance I ran, right?
>
> Teacher: Exactly. So you have 3t=D. Now, Emily runs faster than you,
> right?
>
> Dan: yeah, she is so fast! she runs 4mph.
>
> Teacher: so what is the way to write it for Emily?
>
> Dan: 4t=D [Dan recognizes patterns quickly]
>
> Teacher: great. So you see, t represent something we don't know and
> can be anything.
>
> Dan: what do you mean "anything"?
>
> Teacher: Well, it could be any value but we don't know what it is
> yet.
> If we knew the distance, we would be able to find the time you ran
> [and would go and explain how to, and let's assume Dan has no problem
> with that part].
>
> Dan:But couldn't the speed by any value?
>
> Teacher: not really. Can it be 5mph?
>
> Dan: sure it can. If Blaze is running. He is SOO fast.
>
> Teacher: ok, but it would be fixed for a specific kid, right?
>
> Dan: sure. But we don't know the kid, right?
>
> Teacher: well, in a way we don't.
>
> Dan: so isn't the speed also a variable? why did you call it a
> parameter?
>
> Teacher: because if we knew the kid, we would know the speed.
>
> Dan: but you called "time" a variable, and if we knew the time, we'd
> know t, right?
>
> Teacher: yes. But the time can be anything.
>
> Dan: how can it? we either ran 30 minutes, or 1 hour, or 10 minutes.
> What do you mean "anything"?
>
> Teacher: well, true, but we don't know what it is yet. We have to
> find
> it out first.
>
> Dan: but don't we have to find out the speed as well?
>
> Teacher: If we know who ran, we know the speed.
>
> Dan: ok, so we have to find out how fast each of us runs, right?
>
> Teacher: True, but we already did it.
>
> Dan: right. So how do we find the time?
>
> Teacher: Let's say we know you ran 1 mile. Then...[and shows how to
> solve]
>
> Dan: See? the time is 20 minutes. It's not "anything"! It's fixed,
> exactly like you said about "parameter"
>
>
> I'll stop this conversation here, it could go for an undetermined
> time
> ;-)
>
>
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