On Monday, October 21, 2002, I was on a panel at the annual conference of the Association of Independent Maryland Schools. There were 130 math teachers from private schools in the audience. Here is a list of the questions for the panel and my personal responses. Feel free to do anything you want with them. They are also posted on my web page. http://www.math.jhu.edu/~wsw/ED The format of the panel was: two minutes of opening remarks, each panelist had a special question (I took on the first one by choice) for which they had 4 minutes, each could talk for 2 minutes on the other questions. There was a very short question and answer period. There is no record of what others said but I would like to mention one thing. One of the other panelist is the principal of a major boy's private school in Baltimore. Before being the Principal of the Upper School he was the Academic Dean and before that Head of the Mathematics Department. He said something which I found truly astonishing. He thinks so highly of technology (i.e. he has made his students so dependent on calculators) that he tells his students that when they visit colleges they should interview the chair of the mathematics department and find out if they allow the use of calculators. If they don't then they should consider the college to not be a good fit. Steve W. Stephen Wilson (410) 516-7413 Department of Mathematics FAX (410) 516-5549 Johns Hopkins University Baltimore, MD 21218 wsw@math.jhu.edu http://www.math.jhu.edu/~wsw/ Opening remarks: I am very pleased to be here. It gives me the opportunity to thank you all for the job you do. We entrust our children to you and you have the awesome responsibility of educating them. I consider your job the most important in the country. However, I suspect that this doesn't show up in your pay checks. I want to express my personal profound gratitude and appreciation for what you do. Thank you. Unfortunately, I also want to explain that your job is harder than you are perhaps aware. When you are done with your students you send them to me. I regularly teach freshman calculus in college. Regrettably though, I teach it to about 200 at a time. I can convey a love of mathematics, an excitement about the Calculus and a real desire that they learn the material; but I do not get to know their names or how they think, and I cannot really claim to teach them anything. I run the material by them at high speed in a large lecture and then I examine and grade them. If they survive, it is because you prepared them properly, not because of anything I do. I apologize if I add to your burden, but it is just the way things are. 1.To what extent does the autonomy of each division and its particular mission get in the way of a coherent math curriculum across the grades? Who should be determining the curriculum? I cannot address this issue for all of K-12. However, I can deal with it at the college level. Almost all students in college who are taking mathematics courses are doing so because their major requires it. Their majors not only specify the math course which they must take but the material covered in the course. Mathematics departments in college are service departments, we teach for other departments. Even when we teach for mathematics majors, the material is well defined and not optional. If we fail to teach our students what is required then we hear about it and lots of pressure comes to bear on the department for us to get back in line. If, for example our department (or our University) decided that premedical students should learn statistics instead of calculus, then none of our students would get into medical school (which would be a disaster for Johns Hopkins). The bottom line is: we, in college, have NO autonomy. The end user determines the curriculum. Because my 200 Calculus students have just come from high school I can say something about the autonomy of high school programs. Mathematics professors are in fair agreement about what we need students to know when they come to college. Because Johns Hopkins University is a fairly selective school, if a student is not prepared then they just don't get in. So, high school's have, from my point of view as a teacher at Johns Hopkins, only two choices: prepare the student, or they won't get in. For those colleges which are not so selective, most will give placement tests to entering students. Last fall, in all 4 year colleges and universities in the US, there were 676,000 students taking courses before calculus, including arithmetic. Only 440,000 were in calculus. These numbers do not include community colleges where the ratio is even higher. Very few of the students in the first group will ever complete calculus. It was extraordinarily difficult to get numbers to demonstrate that fact. I finally found one large state university which had them. Of its over 5,000 recent students who had placed into college algebra (material they could have learned in high school) only 7.5% of them ever successfully completed 1 semester of calculus. The lesson is that if you don't prepare a student for college level mathematics before they go to college, then it is very unlikely that they will ever successfully take college level mathematics, thus excluding from their life-options many many careers. So again, the high school autonomy really isn't there either. There are only two options: prepare the student for college mathematics, or they will most likely never take college mathematics. In an attempt to answer this question for everyone, it seems to me that curricula should be pretty much determined by the requirements of the next stage where you are sending students. That's certainly the way it works in college. 2.What set of skills and mathematical dispositions should students have as they leave high school and head to college? First: Entering freshmen need to know how to add, subtract, multiply and divide. They need to be able to do this quickly, easily, and thoughtlessly for numbers, fractions, decimals and polynomials. There is no better preparation for the next step in mathematics than an intimate familiarity with our number system. If a student cannot add or divide fractions and do long division quickly and easily then they do not have the necessary depth to move on to the next stage. Second: Students need to be able to solve complex multi-step word problems. I'll put a good lower school example from a French text up on the screen in just a minute. Lily and Vincent divided a cake proportionally to their ages. Lily was six years old, and Vincent was four. Lily ate one third of her share for lunch, and three fourths of the remainder at dinner. She gave the rest to their cat. Vincent ate three eights of his share at lunch, and five sixths of the remainder at dinner. He gave the rest to their dog. Which had more of the cake, the dog or the cat? Third: And this is harder to grasp because it doesn't affect admissions or placement, but is extremely important. I wish my students were capable of picking up a textbook and learning the material on their own. After all, that's what they have to do in most college math courses. I feel I was very lucky to have grown up in Kansas where the nearest person who knew calculus was 25 miles away. I had to teach myself calculus from a text, without help. When I went to college, I was ready. 3.Many schools report conflict in the transitions between divisions over computational skills? To what extent is the mastery of computational skills, specifically the traditional algorithms, critical to future success in mathematics? The traditional algorithms are the only collection of serious mathematical theorems which can be taught to lower school students. These theorems solve the age old problem of how to do basic computations without having to use different strategies for different numbers. If taught with enthusiasm and admiration, a student should find them exciting and appreciate the awesome power they give. I have already stressed the necessity for proficiency with basic calculations. This can be achieved with any reasonable algorithm, so why insist on the traditional algorithm? My interest in K-12 education has made me pay much more attention to my college teaching and I find that I am constantly using the traditional long division algorithm to solve problems for my students from freshmen calculus to senior math major courses. It is true I am using the long division algorithm not usually with numbers but with polynomials or even in more abstract settings. The traditional long division algorithm easily generalizes to many different situations. If a student doesn't use the traditional algorithm then I can't talk to them and they can't talk to their peers. If you don't teach them the traditional algorithms, then when they get to college they will be isolated from the human interaction which is so very important to learning. 5.Many middle schools are beginning to use graphing calculators as early as sixth grade and yet high school teachers are hearing from returning college students that many college professors are banning graphing calculators. Although the SAT permits their use, the GRE does not. Given the uncertainty of calculator use, what advice do you have for K-12 teachers? Calculators are an important tool and students should certainly learn how to use them. My advice is, however, that a student should always do a calculation by hand if possible. Every bit of practice matters. I asked a cross section of mathematicians whether they agree or disagree with the following statement: In order to succeed at freshmen mathematics at my college/university, it is important to have knowledge of and facility with basic arithmetic algorithms, e.g. multiplication, division, fractions, decimals, and algebra, (without having to rely on a calculator). I got 93 positive responses and no one disagreed. This is particularly remarkable because if you ask this same bunch what mathematics is, then no two of them will agree. There are at least a couple of dozen who would normally disagree just to be disagreeable. Personally, I don't know anyone who allows the use of calculators on exams in college. 4.Some critics have argued that in attempt to engage students the new curricula have sacrificed mathematical rigor. They feel that the repetition once relied upon to internalize procedures is being ignored in the interest of conceptual development. Comment on the relative merits of traditional instruction, constructivism, and any other approaches to teaching mathematics. I am in no position to tell you how you should teach. If I have anything to offer on this panel it is to let you know what a prepared college bound student is and also to let you know what happens to unprepared college students. Unprepared college students don't go on to careers in medicine, they don't go to business school, they don't study science or engineering or computers. They have to do something else. They don't have a choice. They are not prepared. All of us who worry about what to teach and how to teach it are caught on a bit of a treadmill. I have to produce students with certain skills and knowledge. In order to do that I have to get students with certain skills and knowledge. All of us are in this same situation. The curriculum is built, to a large extent, from the top down. To be honest, I don't know where it starts up there, I just know my little part on the treadmill. Do whatever works for you and your students. Just don't try to redefine mathematics and teach skills and knowledge to a student which will not allow the student to succeed at the next level. W. Stephen Wilson (410) 516-7413 Department of Mathematics FAX (410) 516-5549 Johns Hopkins University Baltimore, MD 21218 wsw@math.jhu.edu http://www.math.jhu.edu/~wsw/