Question 1: I am going to answer this question with the assumption that I am talking about all high school graduates, not just those who are planning on going to college. What follows is what I would want for all high school graduates. I would of course expect more from those who are university bound.

I think all high school graduates should have some basic understanding of arithmetic and mathematics. I would want them to be able to compute with fractions, decimals, and percents (with a calculator if necessary) and be able to apply these skills to everyday activities encountered by typical consumers and citizens. I would want them to be able to see patterns and to be able to generalize from patterns and use their generalizations to make sound decisions. I would want them to be comfortable enough with algebraic expressions so that they can use formulas and evaluate expressions when necessary. I would want them to have a basic understanding of plane and solid geometry so that they can handle some of the common consumer problems of buying the appropriate amount of paint, of carpeting a room, or pouring concrete, etc. I would want high school graduates to be proficient with working with different kinds of units of measures. I would want high school graduates to know enough about statistics and probability to read and understand newspaper articles and make good decisions when statistics are given or probabilities are involved. Most importantly, I would want all high school graduates to be able to think and be able to problem-solve. I would want them to have a ``toolbox'' of problem-solving strategies which they can draw from often. I would want them to be able to share their process for thinking or solving problems so that others would be able to understand.

This of course is not a finite list of what I would want high school graduates to understand. These are the things that I care about most.

Question 2: Every sound mathematics curriculum should have the following:

All students should have experiences with all of the strands of mathematics.

The curriculum should be problem-based, where students are presented with a problem and then the mathematics developed to solve the problem. These are not just one-step problems. These should be BIG problems which resemble everyday life and might take several weeks to solve. Mathematics should be relevant to students' lives, and problem solving and critical thinking should be emphasized.

Students should be given opportunities to explore situations, develop generalizations, and construct their own meaning to mathematical concepts before the teacher provides rules, algorithms, and conventions.

Instruction should be balanced between individual and cooperative/collaborative opportunities.

Students should be provided with a myriad of mathematical tools to help them explore problem situations. These could range from beans, pattern blocks, and other such manipulatives to age-appropriate calculators and computers.

Communication of mathematical ideas, in both oral and written form, should be stressed.

Students must have a variety of diverse means for showing what they have learned. Assessment must move beyond paper-and-pencil tests. Assessment must also include observations, oral presentations, written presentations of problem-solving processes, task analysis, project completions, peer evaluations, self-evaluations, and portfolios.

Question 3: If all students of the high school population are completing the core mathematics sequence with passing grades, regardless of ethnicity, gender, or prior math experience, then the program is working well. If students are enjoying mathematics, seeing mathematics as important and relevant, and are considering taking more mathematics than what is required, then the program is working even better. I am continually suspicious when I see mathematics programs which allow less able students to complete their high school math requirement by taking only general arithmetic courses and which never force them to get into the mainstream core mathematics sequence. I am suspicious when I see girls and minorities overwhelming some math classes, while predominantly white males are the ones in the higher math classes.

Question 4: High school mathematics teachers must have a strong mathematics background. More importantly, teachers must truly believe that all students are capable of learning mathematics and being successful. Some students just need more support and time than others, and it is up to high school mathematics teachers and the rest of the high school administration and staff to develop strategies to support those students with more time and assistance. Pedagogically, high school mathematics teachers cannot tie themselves down to a single style. Because our student population is so diverse and because we must meet a myriad of learning styles, high school mathematics teachers must be well versed in numerous instructional strategies. A teacher has to be willing to ask good questions of her students, and then she must carefully listen to their responses. By analyzing their responses, a good teacher should be able to pull from her extensive mathematical background and her experience with diverse instructional strategies in order to present to her students activities and exercises to meet their needs. A good teacher should use discovery activities, experiments and projects, class discussions, questioning, and direct instruction whenever each method is most appropriate.

Question 5: For one thing, I was good at it. But I also saw mathematics as something that was elegant and precise. You could, for example, prove things in mathematics, whereas in other subjects you could only hypothesize and speculate. If you completed a math problem by showing all your work and getting the correct solution, it was considered ``good''. On the contrary, I could work on an English essay or a history paper until I was ``blue in the face'' and my teacher could always find some way to make it better, clearer, more poignant, or something else as obscure. Mathematics has its own rules which are not dependent on the subjectivity of a teacher.

Question 6: First of all, I would want colleges and universities to join the rest of the mathematics education community in implementing instructional strategies which learning theory, educational psychology, and brain research have supported as more successful for learners of all kinds. In my opinion, it is criminal to teach college calculus and not allow graphing calculators in the classroom. It is pointless to present university mathematics students with rules, algorithms, and theory without developing some understanding for such. It is as important at the college level as the high school level to expose students to actual careers and professions where higher mathematics is used and allow students to experience such applications. These would be some of my dreams for my students as they go on to colleges and universities.

For prospective mathematics teachers, once again they need a strong mathematical background in all the strands of mathematics: patterns and functions, algebra, geometry, statistics and probability, discrete mathematics, measurement, number, and logic and language. They need to research how students learn along with the different educational theories. They need to experience a wide range of instructional strategies. They should be comfortable working with groups, providing rich problem-solving situations, using manipulative, conducting student-centered discussions, using rubrics and holistic grading, and utilizing alternative forms of assessment. Wouldn't be nice if the colleges and universities which conduct teacher-preparation programs could incorporate these facets into their training regimes?