(A)Abstraction and (B) analysis for the sake of analysis have characterized the treatment of mathematics in the classroom. Clearly, abstraction and analytic methods are not bad things! This is, after all, what mathematics is all about: ways to generalize problems so that they submit to methods of analysis we have developed. This is the power of mathematics!
The difficulty is that a very large proportion of students need something to help them relate to the abstractions. Many need to know a little about some problem that could be solved before they can successfully focus on learning the analytic methods. Others need concrete and visual support to understand the abstractions.
We have tried to provide two kinds of approaches in the units of elementary mathematics you will find here to help those students.
(1) Some of the units contain activities that use concrete objects to help students develop an understanding of the very basic concepts of algebra, such as variable, equations, and the basic operations. We believe that students who have been unsuccessful with elementary algebra have probably not understood its foundations. This is an effort to provide experiences on which they can build those foundations.
(2) Most of the units are investigations in contexts that we believe students will consider important, contexts that will demonstrate the utility of even elementary mathematics, at the level they are now. Here, we are operating on the assumption that many intelligent people do not learn mathematics because they do not see its connection to their world. We have tried to offer connections in areas that we think will hold universal interest as well as being of special significance for some in future professions.
Thus, where we could, we have tried to offer assistance to concrete and visual learners. More importantly, we think, we have tried to develop lessons that will attract the considerable strength of holistic thinkers/learners by demonstrating context in which mathematical methods make an impact even as students are developing abstractions that can be applied to a wide variety of problem contexts.
Finally, we have designed these lessons for students' collaborative use. We believe that this meets another significant need of many learners. For many, the purpose of learning is social. For others, a social support network for learning is invaluable. Thus, not only does the context of many of these lessons point to the usefulness of mathematics for solving human problems, but the delivery of them offers opportunity for human interaction. Both of these circumstances are expected to support the needs of social learners.
Student collaboration should also serve to take the focus of the classroom activity to the students and away from the instructor. This way, the instructor can facilitate learning instead of merely providing students with recipes. And students can do mathematics instead of merely watching it.