On Nov 1, 2012, at 11:01 AM, Louis Talman <email@example.com> wrote:
> You've devised a thought experiment that demonstrates that you use the phrase "number sense" in a different way than Dehaene uses. It demonstrates nothing about humans' failure to connect what you call "quantitative sense" with the algorithms of arithmetic.
That is because I do not believe they are connected in the way Dehaene is suggesting. I'll get to your modification shortly.
Dehaene suggests that quantitative sense is somehow the natural basis of number sense (regardless of your version, he does state this). I state that quantitative sense is no more the natural basis of number sense than the ability to speak and acquire a language is the basis of reading and writing.
Definition: Quantitative sense is the ability to physically sense quantity. Definition: Number sense begins with the ability to count cardinally.
Speaking and the ability to acquire a language is innate. Agreed? Reading and writing that language is not. Agreed?
If we were unable to speak and acquire a language then obviously we wouldn't go through the trouble of developing reading and writing. Likewise, if we didn't have a sense of quantity, there would be no purpose to developing a sense of number. But just because something is the motivation for something else doesn't mean it is the natural basis of that something else. If we didn't get sick then there would never have been a point to developing the science of medicine (read that as - it would not have happened). But getting sick is not the basis of medicine. Developing a theory of getting sick is the basis of medicine. Likewise, developing a theory of quantity is the basis of mathematics. These are abstract creations to describe real things. They are not the result of those things or some sort of continuation of those things. Animals get sick, but they do not develop a theory of sick. Animals have quantitative sense but they do not develop a theory of quantity. They never will.
Now, your version (and please, correct or add to this as you see fit).
You are saying that students fail to recognize that mathematics is the theory of quantity, a sense they already have and use.
I think they get that. I mean after you tell them and show them. I also think they know that reading and writing has something to do with the language they speak every day. According to previous statements by you, you understood that learning how to play a musical instrument is the path to playing a musical instrument and that you would enjoy playing a musical instrument right now (who wouldn't?). But you didn't do it. Why? My son doesn't particularly love math. If I was out of the equation, he would just get by, which is saying very little by FCAT standards. Now (4th grade) he is showing more interest than dislike but that occurs now because of his achievement, which I had to push him through. Unlike in the beginning where it seems like math (and reading) is nothing but work, now he can enjoy understanding in equal parts with the work. No pain, no gain, right?
I have nothing against real world problems. I majored in physics. How much more real world does mathematics get than that? What I have a problem with is when the course is (allegedly) about real world problems yet there is no mathematical development. No theory of quantity. I especially have a problem when the title of the course is "Algebra". It never bothered me before when they titled the course "Math for Daily Living".
Take a look at that algebra book I posted. It is from the "active learning" camp. One of the first, if not the first, that I have seen that legitimately tries to combine algebra with a more active approach, rather than replace algebra with "whatever you call what it is they do once there is no algebra left in it". (see Discovering Nothing, Dy/Dan, ATTC, etc.). The only point I will make, and I still have to review it more, is that it doesn't really launch the student to our level at any point. But that might be right for the student this book targets. That may be the place to draw the line between algebra and honors algebra. We can't have it all, all of the time.