On Dec 2, 2012, at 11:28 PM, "Clyde Greeno @ MALEI" <greeno@malei.org> wrote:

The concept of  lines of numbers [linearly ordered systems of numbers] was invoked long before Wallis. In accord with the 17th century confusion about signed numbers during his times, Wallis proffered a pedagogically *horrible* interpretation ... the kind of thing that presently abounds among mathematically challenged  educators who have not studied how/what students actually learn.

I must point out for the readers that you mean how "failing students" learn (or, as the case is, don't learn). Some very mathematically and logically proficient readers might take what you write to mean "how all students learn". Your viewpoint is centered on nascent levels of failure in the game (of math), which I somewhat agree with. However, you problem is that you are comparing these failures to later stages of the game. For example, you might walk into a chess class on strategy where the students are failing to understand the elements of strategy and realize that they do not even understand the moves allowed by each piece! So, you would be very right to point out that these students don't even understand the basics. But very wrong to attribute the fault to a class on strategy.

If you think you can cure those nascent failures then by all means, do so. And god bless you. But don't expect us to believe that you have cured anything at all if the student can still not succeed in the classes that follow those nascent beginnings. In other words, we have all been in traditional classes. Telling us that your theory requires us to recant our successes and understanding of our own learning in those curriculums is like shooting yourself in both feet.

Bob Hansen