This is an open letter to the Friends School of Baltimore community about the mathematics program. Please feel free to copy it and distribute it. This letter, a letter about my concerns about the lower school TERC math program and the results of a quick survey of the attitudes of college professors are all on my web site at http://www.math.jhu.edu/~wsw/ED. Dear Friends School: First, I want to express my gratitude to the teachers at Friends School for their dedication and devotion to our children. Friends School is an extraordinary place. The teachers are the foundation of Friends School. They are the ones who touch our children on a daily basis and they are the ones we are most indebted to in our community. I thank them all for making Friends School the wonderful place it is. I have faith in the teachers at Friends School. I trust their competence to teach and their desire to do what is right by our children. I am writing because I believe there is a minor misunderstanding which has serious consequences for our children. I think the curriculum planners at Friends School have been misinformed about the importance of certain aspects of mathematics preparation. I do not believe that the root of this misunderstanding is at Friends School. On the contrary, there is a conflict going on at the national level. The highest level mathematics educators in the country have pushed hard for an emphasis on ``understanding'' and problem solving, at the expense of computational skills and substance. Any reasonable institution would listen to this advice and act on it. Friends has done so, with consummate skill it has emphasized problem solving (at the expense of computational skills and substance). I want to emphasize here that I am well aware of the exceptional success Friends School has with teaching conceptual understanding of mathematics. Under no circumstances would I want Friends to back off from this incredible achievement! So, what is the problem? Where is the misunderstanding? The mathematics educators educate mathematics educators. They are not the people who think about mathematics, who do mathematics, and who teach mathematics to our children when our children get to college. They teach the people who will teach children in K-12 how to teach mathematics. There is a huge gap between these two groups of people: people who teach mathematics education in college and people who teach mathematics in college. It is natural for someone who teaches mathematics in K-12 to look to the mathematics educators for advice as to how to teach mathematics and what to emphasize. My premise is that the source of this misunderstanding is the lack of communication between the mathematics educators and the people who actually teach mathematics at the college level. People who teach our children mathematics at the college level are pretty unanimous about the importance of computational skills. (See http://www.math.jhu.edu/~wsw/list) I want to quickly assert that they are not ``drill and kill'' advocates but ``thrill of skill'' advocates. They support the sort of understanding which Friends School teachers know how to impart to students but they want that understanding to apply towards the traditional algorithms and basic computational skills as well as to general problem solving skills. One can fairly ask: why do they want students to show up in college understanding and proficient at basic computational skills? We have calculators, surely if they know what division means they don't actually have to be able to do it quickly and accurately? When college mathematics professors try to help students move on to the next level of understanding in mathematics they have found that a deep understanding of the number system, an ease of manipulation with it, a comfort with all the operations, is the most helpful prerequisite to moving on and up. You cannot understand mathematics if you cannot do it. Does this mean that we actually have them doing long division on timed tests on a regular basis in Freshmen mathematics courses? No, but those skills make it most likely that the student can absorb the next, more difficult, concepts we will throw at them. [As an aside, students will be expected to know all of the basic arithmetic operations (addition, subtraction, multiplication, and division with fractions) of polynomials (i.e. algebra) in order to move forward. A comfortable working familiarity with this is impossible if a student does not have similar skills with numbers.] So, who should we believe, people who teach us how to teach in K-12 or people who will teach our children in college? The answer, of course, is both. Students should have problem solving skills and computational skills. I would like to put computational skills into a context which would allow Friends school to teach them without any change of philosophy whatsoever. Friends School is fabulous at teaching students how to solve big complex problem. Well, here is one of great importance. It took civilization hundreds of years to solve this problem even after the introduction of Arabic numerals. The problem is: invent a system of hand computation which fits all cases, one which does not require different strategies for different numbers. Create algorithms which allow us to compute arbitrary addition, subtraction, multiplication and division problems without having to think through the problem yet again. Make it work for decimals and fractions and make sure it easily generalizes to polynomials. The truth is that computational skills are the solution to a significant problem: how to compute! Good computational skills are the result of having solved a good problem. Let me make yet another case for the importance of the ``traditional algorithms'' being taught and understood. This time I will take the viewpoint of the pure mathematician I am. The traditional algorithms which solve the problem described in the previous paragraph constitute a collection of THEOREMS of awesome power. The Theorems are incredibly useful. It is true mathematics. These results are probably the only examples of great mathematics which you can show to and explain to a lower school student. They are the culmination of years of sustained study of numbers by generations of very smart people. Don't dismiss them lightly. When a student has understood and developed proficiency with these algorithms then they can do any sort of calculation with ease and understanding without having to rethink a strategy. I have no objection to the constructivist approach of letting a child develop their own strategy. At the end of the day though, the child should be enlightened and exposed to the traditional strategies which have been distilled down to a truly elegant form of mathematics. The point is that these strategies are fabulous, major mathematical accomplishments and they are the only such mathematical results which can be taught to small children. This is their mathematical cultural heritage. One should no more dismiss it than one should dismiss all books written before 1990. Furthermore, viewed properly, these traditional algorithms are incredibly exciting and should be taught as such with great enthusiasm for the very deep mathematics which they represent. I would never presume to know when things should be taught to our children. I trust the faculty to deal with concrete problems like that. Perhaps the secret to long division should be handed out with graduation diplomas? Frankly, I think that awesome power should be shared earlier. My guess would be a few minutes every day from grades 1 through 12 and anyone would become both knowledgeable and proficient. Again, my guess is that the basics should be taught in lower school and constantly reinforced in middle school. Although the use of calculators should be taught, dependency on them should be avoided at all costs (i.e. they should be used rarely). I have tried to be upbeat in my assessment of this problem. However, it is time to emphasize the serious consequences of this misunderstanding. The issue is simple: our children will not do well in mathematics in college without these fundamental computational skills, and this will close off many occupational opportunities to our children. When the word gets out, Friends School will be known as a liberal arts school and our children will have difficulty getting into certain types of colleges. Actually, that is not the real future. What will really happen is that parents will hire tutors for their children when they fully understand what is happening. This will prevent the dire description I have given from coming true. However, Friends School will become (and is presently rapidly heading there) a wonderful school to send your child but where you must (and you will) get outside tutoring in mathematics for your child so your child can successfully deal with the post Friends School world. Is that really what we want of our school? The one thing I know, as a professional mathematics teacher at the college level for 30 years: a student without these skills is unlikely to do well in the Freshmen mathematics courses taught by myself and my colleagues across the country. While I'm not being upbeat, there is need to discuss triage. A fair number of students have survived mathematics at Friends School without learning these important computational skills. For their sake, these students must be tracked down and fixed, so to speak. Sincerely, W. Stephen Wilson Parent of Saul Wilson 4B Postscript: On Monday, March 4, I spent most of the day in Washington attending a debate between pro-arithmetic activists and two presidents (past and present) of the National Council of Teachers of Mathematics (NCTM). The above letter was already written and I must retract my description of the lack of communication between top level educators and people who teach mathematics in college. What I describe is not actually the case. The NCTM is responsible for the standards which led to the Friends School type programs. The present president of the NCTM stated loud and clear that they were not interested in the bright students (i.e. Friends School students) but they were interested in the students who didn't use to learn math. Friends School is teaching with programs consciously developed for students who did not, in the past, learn arithmetic. None of these students go to Friends School. The reason why their standards support the use of calculators and describe long division as ``obsolete'' is because their target students are not going to go to college. After the debate there was a diner attended by about 35 nationally known pro-arithmetic activists. Each of us told the story of his or her involvement. It was actually quite depressing. Aside from California there were few success stories. It became clear that if our concerns about the curriculum at Friends School are actually heard and acted on it would be because Friends School is a truly extraordinary place. I came away quite depressed. It didn't help that during the debate, right on C-SPAN TV, the head of the National Science Foundation (who was in the audience two people down from me) stood up and said she and her husband had tutored their kids in mathematics. It became clear to me then: when confronted with a curriculum like Friends School's curriculum in mathematics, any parent who wants their child to go on to college should get a tutor in mathematics and continue with the tutor until the tutor says ``enough'' or until the child says ``but mom, dad, we do this in school''. My first response was to tell all parents this but I have a conflict of interest. My son and I enjoy doing mathematics together and if the school takes up teaching it then it could usurp my roll with my son. Although I will persist at trying to get Friends School to incorporate an understanding of the basic operations of arithmetic and a good proficiency with their use, I will do so with a light touch. Friends is a very special place and the lack of an adequate mathematics curriculum does not alter that fact. Post-postscript: A 6-th grade parent recently forced me to look at her kid's regular mathematics work in the 6-th grade. I was horrified! It is highly unlikely that a school which approves of a program like that can possibly be made to believe in the necessity to change to an appropriate college preparatory program. Fortunately we are a wealthy country and can continue to import properly educated people from China and India and so, to some extent, change is unnecessary at Friends School.