I very much agree that the tension between language, thought, and mathematics is a key issue [see my word problems book excerpt to what I view as a way to overcome it].
I would (counterfactually? ;-)] be careful about using counterfactuals with students. The reason is exactly because of the problem I believe many have with it (illustrated in the example below). The reason is that counterfactuals require more mental processes than factual language. One has to hold both states in his mind when one is using counterfactuals. It's more accurate mathematically (logic wise) but more complex thinking wise.
Thus, I prefer them to simplify the language and then translate to mathematics.
At the same time, it hink logic and learning to apply it is a CRITICAL thing that should, but is not, studied in schools, and should be studied early. Yeah, they study ven diagrams but not in a way that provides them with the ability to apply it in general to domains other than the narrow type of problems they see when learning it.
By the way, I probably were not careful with my language in the "example of a conversation" because I was writing it on the spot, quickly, and to an educated audience that can pick out the mistakes ;-) [of course, some of those by the student are certainly on purpose to imitate students' thinking].
Just one word about "learning abilities" studies (and I don't mean to open a whole discussion, although we could on a different thread): how can you divest the ability of the student to learn from the way s/he is taught without doing a VERY big study across MANY learning methods?
And I'm not even bringing up (which of course I am doing) the students' past education.
And yet another point: learning is very much a "brain resources" issue, in the sense that if we do something a lot and intensely, our ability to learn it increases (assuming we are not practicing mistakes - hence the futility of drill for the student who doesn't understand). Abstract thinking is similar. The problem is that students are expected to do it on & off and have long breaks in between. Therefore, only those who are naturally so inclined (and usually do it all the time, or most of the time, anyway) do well.
--- In email@example.com, "Paul A. Tanner III" <uprho@...> wrote: > > 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@...> 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 > > ;-) > > > > >