On Oct 31, 2012, at 1:32 PM, Louis Talman <talmanl@gmail.com> wrote:

I think Dehaene makes a good *experimental* case for an innate quantitative sense, to use a phrase you prefer, in all higher animals and many lower ones.

Agreed.

And the hypothesis that "mathy" kids are the ones who manage to connect that innate sense with the algorithms of arithmetic, while one cause of non-mathiness is failure to do so, offers some explanation of things I've seen in the classroom.

There are kids who don't know when addition is appropriate or when multiplication is. That seems to me to be well explained by such a failure to connect.  This is exactly what one would expect of a kid who learns algorithms because required to do so instead of as something connected to the real world.

First, a small matter of business. I will PayPal \$100 to the first person that sends me an elementary school textbook or produces a verifiable copy (online or whatever, I must be able to see the whole book) of an elementary school textbook, that was published and in use by schools in the last 60 years, that teaches addition and subtraction devoid of use cases. In other words, it must teach addition and subtraction entirely as rote mechanical algorithms. I don't care if it is the standard algorithm or some other algorithm, but it must be devoid of application. No pictures of 5 apples and 3 apples and no word problems like "John has 10 apples and gives Mary 3 of them, how many does he left?"

Sorry. I had to get that off my chest. I realize that even if there is no such textbook a really bad teacher could still teach as you describe. It would be one hell of an accomplishment. I just want to prove that such a method of teaching isn't according to any established method I am familiar with, traditional or otherwise.

So, back to what you were saying. In order to know when to add or subtract you need to know how to add and subtract, correct? So, whether you know just "how" to add and subtract, or both "how and when" to add and subtract, you have to know "how". To know "how" to add and subtract you must understand numbers. When I ask my son "What is the smallest number greater than zero?" and he answers "There isn't one.", by my definition that is number sense, not quantitative sense. He does not get that sense all at once, he gets it over time, by working with numbers. That all begins with counting, adding and subtracting.

Imagine the following experiment...

A random number of objects, less than 12, and all of the same type are placed in a random arrangement on a table.
A random number of objects, greater than 20, and all of the same type as the first are placed in a random arrangement on a second table, next to the first.
The task is to move the correct number of objects from the second table to the first in order to have 12 objects on the first.

If a chimp could do this repeatedly, then I would concede number sense.

Many here might claim that I am asking the chimp to do too much. They will claim that I am asking the chimp to subtract. Paul will claim that I am asking the chimp to solve y = 12 - x. But if you have ever taught a child arithmetic you would know exactly what I am doing. I am proving that the chimp can count. If there are 8 objects on the first table the first (very famous) chimp that is successful at this task will count 9, 10, 11, 12, as it moves the objects over.

I provide this thought experiment because it is an even more basic example of your example, the student that doesn't know when to add or subtract. It also involves the "real world". It also shows that this has nothing to do with quantitative sense. The reason the chimp cannot do this simple task is that it lacks the ability to count. It lacks the sense of number. I propose that is the same case in your example.

My hypothesis is that quantitative sense and number sense are two very different things. One is concrete and physical while the other is imagined and abstract yet can be applied to the first.

The issue then becomes one of devising *experiments* to follow up on this hypothesis---not, as you seem to think, offering rationalizations for not believing it. Most of those rationalizations can be easily defeated by noting that humans have bigger, more versatile brains---which are capable of extending innate qualities in ways that animal brains aren't.

What couldn't be so defeated is evidence from well-defined experiments.

Well, I have devised the experiment, and solved the riddle, without even having to perform the experiment. There is the line between quantitative sense and number sense.

Bob Hansen