Gaps in developmental math students' knowledge

The Pierce College Modular Math 115 project for a modular, self-paced, mastery-based elementary algebra course is in its third semester.

We've made adjustments each semester.  This semester we've used the open source online homework system WeBWorK so that students can verify that they have the correct answers on all the drill questions, so we no longer collect those homework sets on a daily basis.

This semester we are acknowledging how inadequately we have met the self-paced aspect of the plan.  The four sections of Modular Math 115 all meet at the same hour so that we can physically relocate students to the classroom that is progressing at the pace most appropriate for them.   In order to accommodate students who cannot master our elementary algebra materials within one semester, we are moving the slower students (electronically) to a course in the district database that covers only the first half of a two-semester elementary algebra course.

Among the 43 students in my classroom, none had mastered even two (of the nine total) units by the end of the twelfth week of fifteen weeks in the semester.

The students suffer all the mathematical gaps we have come to expect, such as inability to distinguish the concepts of area and perimeter.  But I was surprised by how many of the students have only a very shallow understanding of subtraction (and of course of multiplication and division).

They are all capable of computing 5 - 3.  And they can easily answer, "If you had 5 pencils and I took away 3, how many pencils would you still have?"

That particular word problem involves the simplest model for subtraction, "take away".  But my students have trouble with the more sophisticated, "If your pencil box holds 5 pencils and you already have 3, how many more pencils do you need to fill the pencil box?" 

The students do not automatically recognize their task as computing a difference.  Instead, they solve such a problem by counting up from 3, and so they have even more trouble with "If you had some pencils and then I gave you three more so that you had a total of 5 pencils, how many pencils did you have at the start?"

And they have more difficulty with a comparison question, "If you have 3 pencils and I have 5, how many more pencils do I have than you?"

Some of the students seems unfamiliar with the idea of multiplication as repeated addition--they know some multiplication facts but do not recognize that one can compute 3+3+3+3 by multiplying 4*3. 

Even if we assume that our students have access to technology to carry out computations and symbolic manipulations, some of these students do not recognize what calculation or manipulation is useful when given a context outside of pure mathematical computation.