> 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. >
I agree, no textbooks with just numbers and no other objects.
I like to sort on whether an abacus is shown or mentioned. That tends to bode well, is one of those early signs we might have a winner.
> 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. >
I would disagree that "to add" and "to subtract" are necessarily symbolic operations. We may subtract wood from a wood pile and add it to the stove, add more water to the soup, take carrots from the garden. Add and subtract have to do with moving objects through distance (translation) in that we subtract them from a "place" and add them to another.
Numbers come in when you're tallying a countable number of like type. By "countable" I don't mean anything fancy, just that pouring rice or sand is not a matter of counting grains. One goes by volume or, better, weight. To speak of tiny "atoms" ala Democritus does not change that fact that we sometimes add in bulk, without precise integer measurements. The game of "roughly" or "approximately" is quite rule-bound, meaning no less a language game than one of integers and abacus adding (with positional notation vis-a-vis a base, a carry operation, and all the rest of it).
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> 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. >
I get the impression the initial hypothesis was some students will fail to connect these two, but maybe not in all contexts.
> > 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. >
Human brains have a way of filling large tomes with all kinds of intra-brain traffic such as spy novels and romances.
We don't really know much about the mental life of other animals and whether these speculations have any utility. What do whales share about?
> 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
If I fetch three logs for the fire, subtracting from the wood pile and adding to the stove, is that quantitative sense or number sense?
If I was asked to "get three logs" and obey, is "number sense" now in the foreground? Is "counting" alone enough to establish "number sense"?
When I see two people each have half a load of hay, or half a loaf of bread, I am seeing that they have copies of "the same thing" (loaf of bread).
I am able to detect greater than, less than, equal, without having any names for units or amounts, correct?