Question 1: I want students to understand that mathematics is a discipline with wide-ranging application to thinking and problem solving. I want them to see mathematics as more than just plug-and-chug algorithmic computation. To me this requires using mathematics outside of math questions. Discussing the role that math plays in physics, chemistry, biology, and other fields. Of course, it is also important that graduates receive basic training in skills necessary for continuing on in mathematics at the college level.

Question 2: There are two conflicting interests. The first is preparing students who will go on in math and science for the college curriculum, and the second is preparing students who are likely to never take another math class. The needs of the students are different, and therefore, the essentials are different.

Things essential for both types of students are (in no particular order):

1. Estimation and ability to check answers.

2. Analysis of problems to see the mathematics beneath them.

3. An aim at understanding how math relates to the world.

4. Basic probability and statistics so that students can understand newspaper articles and science.

5. Other basic skills.

On the collegiate math and science preparation level, the essentials have to include some understanding of important properties of functions and their graphs, such as roots of a polynomial and how to find them in simple cases. Additionally, basic algebra and geometry are a must for preparation. I think, however, that there should be an emphasis on applying the ideas rather than simply pushing symbols around.

For those who will not be continuing on in college mathematics, the needs are only slightly different. For too long, I think we have been teaching symbol manipulation for these students rather than understanding of how to apply the ideas. There should be more contextual problems and problems that show the need to use the mathematics they know and the mathematics others know.

Question 3: This is a much harder question. The best answer is that we have students who are interested in solving math problems themselves rather than by aping the examples done in the book.

Question 4: Understanding that mathematics is more than plug and chug is the most important thing to me. I would like to see them help students learn mathematics on their own.

Question 5: Seeing and solving the "handshake problem" when I was in sixth grade. A friend had been asked the question of given n points in a circle, how many distinct chords can be drawn with the given endpoints. I worked on the problem recognized the pattern 1 + 2 + . . . + (n-1), and then my mother showed me how to turn this into a general argument (and the formula for the sum).