Question 1: I believe that high school graduates should be able to make effective use of a wide variety of mathematics and mathematical methods in situations that may not be familiar to them. They should be able to use mathematical concepts, mathematical methods or tools, and a variety of problem solving strategies, and be able to communicate mathematically using symbols and formulae, graphs or diagrams, numbers, and appropriate conversational language. They should be able to use algebraic and geometric as well as numerical methods as appropriate. For me, the critical goal is that they become thinking users of mathematics rather than mimics of what they have seen others do. They should "figure things out" more often than trying to "remember what they were taught to do." They should be able to explain what they have done, why they have done it, and why it provides the information sought or responds to the question they were trying to answer. They should recognize the power of abstraction across applications. They should seek and recognize patterns or symmetry.

Mathematics is a language that is natural and useful in understanding a wide range of aspects of our life experience. Most of all, mathematics is more a matter of "thinking" than "remembering."

Question 2: A high school mathematics curriculum should present mathematics as an interconnected whole. It should include the study of functions, algebra, geometry, statistics and probability, discrete mathematics, logic, reasoning and communication, applications of mathematics, and numbers (integers, rationals, real and complex numbers, as well as other number systems). Thinking of these as strands of mathematics, a good visual image is of mathematics as a tapestry whose beauty and strength requires the artful interweaving of the strands. Such a high school curriculum is best understood when viewed from a distance so that the unity and proper proportional relationships come into view. The curriculum should be less hierarchical and more multidimensional with "advanced" issues suggested early on and with "elementary" topics revisited frequently at greater and greater depth. The use of credible mathematical models, problem solving (including the development, use and analysis of algorithmic approaches), mathematical communication (in all its manifestations and translations between forms), exploration, conjecturing or guessing, and the development of explanations (arguments or "proofs") should be part of the overall experience. A deep conceptual introduction to the mathematics of change is important, but a formal course in calculus is not necessary. Most importantly, thinking about mathematics and reflection on what has been learned should be central to every student's experiences in high school. Accuracy and depth of understanding are more important than speed in carrying out narrowly prescribed operations.

Question 3: If high school mathematics were working well students entering vocational, apprenticeship, or employment situations would be able to quickly begin learning and understanding mathematical aspects of their work. They would not require retaking courses nor would they avoid opportunities because of a fear of the mathematical sciences. For example, we are told that many situations today require the use of technology which provides them with data presented in numerical or graphical formats. They will be able to interpret and make decisions based on this information as well as understand the limitations that might be inherent in the data. They should be able to make appropriate estimations and to recognize when some mathematics "doesn't make sense." For example, when a calculation has produced an impossible or unlikely answer or when an algebraic or arithmetic procedure is wrong.

For those students going on to technical, college or university study, many of the same criteria are appropriate. In addition, high school mathematics should not be a barrier to participation or success in quantitative courses such as chemistry, physics, biology, statistics, engineering and, of course, mathematics. While cost may continue to require the use of anemic surrogates such as multiple choice examinations, the real criteria for measuring success is the ability to successfully do meaningful larger mathematical tasks. Success in those advanced courses and careers which make significant use of mathematics are the real goals and, therefore, the real measure of how well high school mathematics education is working.

Question 4: Many things are important to successful teaching at any level. Weakness in any area can limit success overall. I believe that teachers at all levels must have a stronger understanding of mathematics in all its manifestations. A healthy attitude about learning throughout one's life and across domains, including mathematics, is an important attribute of a successful teacher. The craft of teaching requires the teacher to be able to use a wide range of teaching strategies according to the context. No single approach will be optimal for all students at all times. For example, no musician learned only from attending lots of concerts and trying to mimic the performers, but every successful musician has watched lots of performances. The key, I believe, is developing all the "teaching" tools and knowing what to use when needed. Coaching or guiding the learning of students by using small group or collaborative learning methods (a controversial approach for some) has been used by outstanding teachers forever. But, these same teachers also "tell" or "lecture" as well as "demonstrate" or "model." What about the attitude of the teacher towards the students? How does this attitude influence student performance? The women students under-recognized in classroom discussions, the changes in the teacher's voice when speaking to students of another ethnicity, the lowered expectations of students from the "poor neighborhood" or who aren't in the "gifted group"--these all have the most profound impacts on the success of the student.

I have been in high school classrooms where the performance of the students was limited by each of these. And there are more ways in which teaching can fail. It is not a question of which is most important. They are all important!

Question 5: I was first attracted to mathematics by my experiences in high school mechanical drawing (a vocational course) and Euclidean geometry. The first course involved figuring out the three dimensional nature of objects from limited visual information and the production of perspective and other drawings. The second course was a traditional "two column" proof course during which I experienced the challenges of finding language to "prove the obvious" and the fun of working with my classmates trying to solve problems for which there might not be a solution or which were several levels more challenging than the standard homework. In both courses, the teachers provided a rich challenging curriculum and an environment that encouraged creative thinking, questioning every detail, as well as being respectful to all students and their efforts. Mistakes were seen as opportunities to learn, not a measure of lack of ability. In mathematics I learned the difference between understanding and not understanding. And understanding, once gained, was forever. There was, of course, the sense of adventure, the exploration of unknown intellectual worlds, the exhilaration of discovery, and the fun of sharing these discoveries with others. I felt a greater opportunity to chart new frontiers. To have something of the experience of being the first human to walk on the moon. Mathematics was, and still is, a grand adventure taking me to unexplored places.