Question 1: I care that high school graduates understand that they can do mathematics, that mathematics is more than algorithmic computation, that many people do mathematics without being mathematicians, and that mathematicians play by many different sets of rules or axioms--not all of which are pertinent in the real world. I would like for high school graduates to understand that they can attack problems in many different ways and most often the answers at which they arrive must be judged based upon the assumptions made at the onset or as decisions arise in the solution of a problem. The solution must also be judged based on whether or not it solved the original problem. I would also like for students to be understand that they will use mathematics in the future, not necessarily traditional algebra or geometry and that mathematics is a holistic subject and not necessarily compartmentalized into the traditional subject headings on high school transcripts.

Question 2: There are serious implications when a question like this is asked. The question is very generic and I think that what you may be asking is about college intending students. I choose not to answer that question right this minute. Let us consider what one needs to be mathematically literate if one has the minimum amount of mathematics in high school. Whatever is identified here is mathematics that all students should know whether or not they choose to go to college. However, the mathematics talked about here is mathematics that many college intending students do not normally get in a traditional secondary mathematics program. That mathematics should include a basic understanding of probability and statistics, especially statistics at a level that graphs and charts in common newspapers--yes even USA Today--can be understood. The probability should be at a level that students can understand odds and probabilities at least at the level that they can understand lotteries and probabilities of baseball players getting hits in at bats. These involve very different types of expectations than those usually discussed. Further students should understand some basic geometry and trigonometry at least of the triangular variety. By basic geometry, I do not mean traditional proofs. I do expect all students to be able to present logical arguments for why they do what they do mathematically. In addition, I feel that every high school graduate should be familiar with technological tools that they will use outside school. The technological tools are just that. They aren't a substitute for mathematics, but students should know that just as word processors made writing somewhat easier, the technology can make some mathematics easier.

Question 3: We would know that high school mathematics is working well if we have industry telling us that their new employees easily adapt to the mathematics that is necessary to do jobs. I would not base how well mathematics education is doing based on how high school graduates do in college or university freshmen classes, because many of those are so antiquated that they are of no consequence in real mathematics. For example, many freshman classes are based upon what I call a junior high mathematics mentality of ten years ago. In the past, junior high mathematics was two more years of doing what you had done for the past six years with almost nothing new. That is the same type of mathematics in most freshmen college classes. Also, the majority of college entrance tests and placement tests cannot be used as the judge and jury to evaluate high school programs. Virtually all of those are based upon algorithmic computation with little understanding of what should be in high school classes. A true evaluative instrument would be one that would pose a real problem that could be attacked with mathematics and technological tools where all assumptions have to be shown and assessment takes into account those assumptions and the logical reasoning used to reach conclusions. Students should be able to write a summary of all the processes used to reach the solution.

Question 4: This is a set of questions, not just one. High school teachers of mathematics should be made aware of mathematics as a field and not just a series of courses. They should have a broad background to include linear algebra, mathematical modeling, axiomatic structures, methods of teaching, probability, statistics, with some analysis and other mathematics. The background should be broad, should include mathematics history, and should use technology in the teaching and learning of it. Prospective high school teachers should learn from their professors and from teachers in the field that mathematics is not just what is printed in textbooks. Mathematics is not simply a set of rules to be followed. They should understand that not all people learn in the same way and should be taught much about how mathematics is learned by people. A course or courses in learning theory is appropriate. Multiple approaches to presentation is most important. Of most importance is the understanding that all people can learn mathematics that is significant and important.

Question 5: I had a fantastic high school mathematics teacher. This woman made me believe that I could do mathematics and that she liked what she was doing. Her name was Mildred Majors, and she taught at Adamsville High School in Adamsville, Tennessee. I have gone back and thanked her for what she did for me.

Question 6: Higher education in mathematics must change to accommodate to meet the needs of mathematics for future generations. Professors must learn how to use technology effectively, must learn how to use different methods of delivery and must learn how people learn. To do any less is unacceptable. Mathematics departments must re-think how classes are taught and what is taught. Calculus is not the basic course for the majority of people and may not even be the needed freshmen course for mathematics majors. Placement tests must be changed. Use of standardized tests must be re-thought. AP placement exams must be re-thought as must AP courses. The time is past when university professors delegate teaching of lower level courses to teaching assistants and when research is more important than teaching.

The same is true in spades for prospective secondary teachers. Teachers of prospective secondary teachers should be well aware of and be able to work with Schools of Education where methods of teaching are taught. There should be a seamless transition between content and pedagogy at the college level. University professors should not look down their noses at their pre-college counterparts and should be cognizant of the fact that many of the pre-college teachers are very good. They should realize that teaching mathematics is not an us versus them situation. WE are part of the problem as Pogo said.

We have very good evidence that traditional programs do not do what we want done. We have good evidence that is growing that different mathematics may be far better for high school students. Can anything based on research be said about what colleges and universities should be doing in mathematics for the majority of students or even our majors? How would our undergraduates stack up in an international study as is done of pre-college students?