Resources for teaching arithmetic to calculus students Every time we teach calculus we discover that a significant portion of our students never understood arithmetic.  I don't mean that they can't multiply numbers, but rather that they don't know intuitively that a car going 15 miles an hour goes 1 mile in 60/15=4 minutes (i.e. that division is the arithmetic operation corresponding to this problem).
It would be entirely inappropriate to teach them as if they were 10 years old, even if we had 3 months to teach them arithmetic.
Usually these are fairly intelligent individuals considering that they managed to get through high school mathematics well enough to get into a good college or university despite this handicap, and this deficiency in their background is not their fault in any way.  They are likely to be able to pick up arithmetic quite quickly, and figure out from that why they have been a little befuddled through all of high school math.
I would hope this problem has been studied and ways to help these students have been proposed.
I am looking for references either to resources for these students or resources for instructors trying to help these students in the context of a calculus (or precalculus) class.
 A: I have TA'ed a "Mathematics for Future Elementary School Teachers" course.  The point of the course is to develop a deep understanding of elementary school math (read: An actual understanding, rather than a knowledge of how to do computations).  The book we used was Sybilla Beckmann's "Mathematics for Elementary Teachers".
At the end of the course, most students could really explain why 2/3 of 4/5 of a cup of milk was 8/15 of a cup of milk, and could draw a picture which showed why it was true.  Ditto for the addition of fractions, and the algorithms for addition, multiplication, and division. I had many students who were flabbergasted that no one had ever shown them why these things were true before.  Of course, I didn't actually show them:  Sybilla's book is geared toward activities which help students to discover why these things work on their own or in small groups.  The role of the teacher is to direct and clarify.
The reason that this course works, though, is because the students (at least initially) think that they are only learning how to explain these things to elementary school students.  You never come right out and say "You do not understand addition, and I am going to show you".  So it is a unique circumstance.  Even then there are many students who resist the course because they feel like they don't have to put in any work to understand such "basic concepts".  A lot of these students turn around when they realize that they do not really understand, and see that they are doing poorly on examinations and homework.  Some of them do not ever feel comfortable enough to face their ignorance, and these people generally do not do so well in the course.  A teacher must be humble enough to realize when they do not understand something, so it is a good thing that this course is a requirement for future teachers.
If you are serious about starting a course focused on elementary school math at the college level, which I think is a GREAT idea, I would use Beckmann's book.  It is really fantastic.  If you want more info, like an actual plan for a quarter's worth of work, I could email one to you.  
