The most difficult task I undertake with each and every new group of students is helping them reconceptualize the nature of mathematics. I have come to believe that an appreciation for what it means to "do mathematics" is critical for teachers to actually engage in Standards-oriented instruction. For most novices, mathematics is numbers and arithmetic, and making change in the grocery store or balancing a checkbook. It is difficult for students with this orientation to the subject to appreciate what this book is about or what Principles and Standards is attempting to tell us.
Traditional Views and New Approaches: Math Makes Sense!
As Magdalene Lampert points out, most people believe that mathematics is associated with certainty; that knowing mathematics means being able to get the right answer - quickly. (See Transparencies T-7 and T-8, adapted from Lampert.) The students in your methods class will likely share these beliefs. They have seen mathematics as a set of rules given by the teacher that the students are expected to follow. Their experiences will likely confirm the idea that correctness is determined by the teacher or the answer book. Unfortunately, many of your students may have come from college mathematics classes where this view of the discipline continues to be fostered.
All students, beginning with your prospective teachers, must come to believe that mathematics does not "come from a math god." They must experience what it means to make sense of a mathematical idea, to engage in discourse within the classroom, and to come to see mathematics as a community endeavor. At least as important is the notion that the validity of a mathematical idea is found in the logic and order of the mathematics itself - not in an answer key.
Another way to say this is MATH MAKES SENSE! See the box on page 14 and also transparency T-10. I hope that every day you help your teacher-students come to believe that math makes sense, that they are capable themselves of making sense of mathematics, and that therefore they can help their students make sense of mathematics.
Mathematics as a Science of Pattern and Order
The definition of mathematics as a science of pattern and order is taken from Everybody Counts (T-9). In recent years I have found it helpful to my own formulation of what mathematics is about to reflect on this idea. That mathematics is termed a science speaks to the investigative nature of the discipline. The notions of pattern and order can be found in nearly every facet of mathematics from counting to fractals. It is a phrase I want to introduce to my students early on and have them hear it throughout the course as a "hook" for much of what is important.
The Role of the Teacher
As you help students see that mathematical ideas can and should be developed by the students and not come ready made from the teacher, the role of the teacher in the mathematics classroom must also change. Teachers are not dispensers of knowledge but facilitators within the mathematical community of the classroom. This is a good place to direct students' attention to the Professional Standards. Refer again to the five shifts in classroom atmosphere that were listed in Chapter One (also T-12). By the end of this chapter you would like for them to have read some or all of the seven standards for teaching mathematics and have a context within which these ideas might make sense. The bullets for these standards can be found in Appendix B, and of course the entire text can be found on the NCTM web site.