Dear Pam, Sorry this is so late but we were away in Mexico visiting my father much of June and I have come back to my office computer with a broken network card so I have to work at home. I am writing, as you requested, about suggestions for texts in mathematics for the lower grades. Friends School students are among the educational elite in the country. They will all go on to college, which means that they will almost all take some mathematics course(s) in college. Many of them will pursue technical careers or manage those who do. I have little knowledge of what students really should be doing in the lower grades in mathematics. My ``expertise'' is on what they must know when they hit college and how well they must know it. I will ignore things like trigonometry and concentrate on things they need to know which I can easily trace back to learning in the lower grades. There are two things which I will focus on. First, students must, of course, have some basic problem solving skills. This should certainly start in the lower grades. Problem solving is actually a well understood, highly structured activity. However, I have seen no books which approach it in this way except for specialized college texts. It seems that the way the lower grades approach problem solving is by supplying increasingly complex word problems. Some texts do this better than others. Some approaches let students reinvent problem solving and others teach by example. A little comment on the side here about creativity: Many people like to emphasize creativity at all levels of education. However, you can get through life quite nicely with good problem solving skills and no creativity. On the other hand, a creative person with no problem solving skills might struggle to survive in our technological society. Good solid problem solving skills are already very rare at the college level where all Friends School students are headed. I would emphasize problem solving over creativity any day. Second, students must know basic arithmetic operations as applied to polynomials. That is, they must be able to add, subtract, multiply, factor, and divide polynomials and their fractions. They must be able to do this quickly (and accurately) without having to think too much about it. Polynomials are the hard case. In order to learn to work with them a student must first learn the basic operations mentioned above with numbers. Thus it is important to learn to use the standard algorithms for these operations, the algorithms which have been developed by society over hundreds of years (keep in mind that we in the ``western world'' have only been using Arabic numerals for about 400 years). These algorithms generalize nicely to polynomials and the student with good facility with numbers will most likely be able to adapt the same skills to polynomials. This facility is gained in the lower grades. Saul, for one, certainly does not have this facility, even for the most elementary operations, after the third grade. (He needs regular timed tests which he now gets at home after I discovered that he was 1/3 the appropriate speed for grade level.) Students who invent their own way to do long division and are not then taught the standard algorithm, might have serious difficulties when it is time to learn to divide polynomials. (In college, say at Johns Hopkins University, a student is in a calculus class with 200 other students. Exams are 50 minutes and contain several calculus problems each of which involves several basic algebra manipulations. The student who has to think about these algebra manipulations never gets to the calculus part of the problem, and does not do well on exams.) So, when I look at texts, I look for these two things: problem solving and basic arithmetic skills. I look at the TERC workbooks when they come home. I find their mathematical content to be on the level of kindergarten. They give students a working familiarity with individual small numbers, a good thing. However, it is easy to tweak any program to do that and it should be done and over with by the time a student gets to the third grade. (Playing the board game Monopoly does this very very well. Just play until you can move the pieces without counting. This is significantly more effective than any of TERC's made up games which have come home with Saul as homework which I have had to play.) One of the fundamental aspects of TERC is its belief that students should develop their own strategies for problem solving. However, you do not let students at Friends School decide for themselves what the structure of a paragraph should be so why allow them to invent how to solve a mathematics problem when there is a well structured approach to problem solving which can be taught like the structure of a written paragraph? I have a number of colleagues, mathematicians and those well educated in mathematics, who are very much involved in mathematics education around the country. They tend to rate the TERC program as the lowest of the new National Council of Teachers of Mathematics (NCTM) standards based mathematics programs which have come out in the last 10 years. See for example: http://www.mathematicallycorrect.com/books2f.htm\ \ They view TERC and similar programs as having been developed to benefit the bottom 25% of the school population. This may, in fact, be a good thing for this bottom 25% of the population but NONE of those students is at Friends School. This view was confirmed, much to my surprise, during my breakfast conversation with Joan Ferrini-Mundy earlier this year. Joan is presently the Chair of the committee which is revising the NCTM standards. She is a Dean and full Professor at Michigan State University now but was, until recently, Director of the Mathematical Sciences Education Board for the National Research Council's Center for Science, Mathematics and Engineering Education in Washington. When I expressed my concerns about how these programs, particularly TERC, prepare students for college mathematics courses, she responded: ``but what about those who don't go on?'' She expressed no interested in the kind of student who goes to Friends School. I have been following the textbook debate for some of the New York City schools where some of my colleagues are quite active. One of the districts which is a major focus of their activity uses TERC. Here is an interesting quote from one of those emails: While it may be true that 76% of District 2 students meet state standards, the dirty little secret is that our scores are skewed because parents are resorting to private tutors. Many parents in schools like PS41, 6 and 234 have the money to pay exorbitant hourly rates for tutors, superseding TERC. The percentiles do not reflect the success of TERC, but rather the financial success of a fairly large portion of the parent body. There are also many parents for whom this is a financial burden, but they have resorted to tutoring out of desperation. If the scores of children who have had private tutoring in traditional math could be factored out, District 2's scores would present an altogether different picture. While Saul was at Park School I attended two mathematics meetings with the local TERC representative who was a very strong advocate that the standard algorithms of basic arithmetic operations should NEVER be shown to students. For the reasons I have already given above this was sufficient to turn me off to TERC. Seeing the workbooks took me the rest of the way. From the same New York City discussion group, more about TERC: One District 2 parent testified to the school board last year that she had inquired at 8 private schools about admission in 5th grade, for her fourth grade daughter, who then was enrolled in a gifted program at PS 11. All 8 private school directors told her they'd found students coming from District 2 schools, trained with TERC, far behind their students. By the way she also found that in her school, ALL the 4th graders in the gifted program, who scored at or above the cut-off on the 4th grade state assessment defining eligibility for the district's middle school special progress programs, had received private tutoring. My colleagues think little of the NCTM standards, the National Science Foundation (NSF) funded new mathematics education programs and many ``education experts''. They recognize how much power and influence these groups have though. Their main concern is that many people involved in these enterprises know little of the mathematics that they are supposedly preparing students for. Historically, the ``education experts'' wrote the NCTM standards in the late 80s and controlled funding in the NSF during the 90s which only supported the development of NCTM standards based mathematics programs, of which TERC is an example. Many of my colleagues involved in mathematics education are in California where these new programs took hold and upset many parents. These colleagues have now completely taken over things in the state of California, rewritten the California Standards, see http://www.cde.ca.gov/board/pdf/math.pdf\ \ , the California curriculums, and control the approval of text books state wide. In addition, like minded people will now control the funding at the NSF for mathematics education programs. They assure me that all of the NCTM Standards based mathematics programs presently being funded will lose their funding when their grants come up for renewal. The history of texts in our household for Saul is as follows. I started off using a workbook from Saxon Publishers: http://www1.saxonpub.com/\ \ . It was fine but then I read some reviews (comparing Sadlier, Saxon, and SRA McGraw-Hill), http://www.mathematicallycorrect.com/k6books.pdf\ \ , and switched to SRA McGraw-Hill, http://www.sra4kids.com/teacher/math.index.html\ \ . I am quite happy with these books although I didn't actually buy the (expensive) texts but only the cheap workbooks. I finally came to the conclusion that I really needed a text so I bought the HIGHLY recommended Singapore series http://www.singaporemath.com/\ \ . (Mathematics is cumulative and one of the serious problems with TERC is the lack of a text. A text is necessary so that a student can go back and review things when s/he needs to.) My colleagues around the country who are concerned with these matters consider Saxon a good solid text with a great track record. In particular they know of many schools, both public and private, which have used Saxon with great success. I hear very little about SRA McGraw-Hill other than the review I read. Everyone considers the Singapore series to be vastly superior to anything produced in the United States. There are a couple of small drawbacks with it though. They spend no time on graphs, only use the metric system, and use Singapore money. On the plus side, they are very short so there is plenty of time for tweaking the program to compensate for these shortcomings. (There are 2 short paperback texts a year and each has 2 workbooks.) I was happy enough with Saxon but happier with the workbooks of SRA McGraw-Hill. The Singapore series is outstanding in the way it presents the basic arithmetic operations (which I think are so important). I will continue to use the Singapore series for this purpose. However, I found the SRA McGraw-Hill series to have more interesting, complex, and thought provoking word problems. Therefore it seems to me to be a better choice (for my home schooling) for teaching problem solving. If I were to recommend one series for Friends School from those I am familiar with I would probably go with the SRA McGraw-Hill series. Of course I would want to look more closely at the actual texts as opposed to just the workbooks. Some other texts which have been approved in California, where I know the folks in charge, are listed at: http://www.cde.ca.gov/cfir/math/2001adpr.pdf\ \ Additional reviews of some of these programs are at: http://www.mathematicallycorrect.com/books.htm\ \ . In general, to see the views of those who do not like the NCTM standards based programs, you might want to browse the site: http://www.mathematicallycorrect.com/\ \ . If you consider changing mathematics programs at Friends School and would like input from me (and my colleagues) about specific texts then I would be more than happy to look closely at anything you like. I also reiterate my offer to spend volunteer time at Friends School doing something with mathematics in any capacity which would help you out. For example, I could meet occasionally with juniors in high school who are interested in mathematics. (Perhaps a letter of reference from me would help them with their college applications.)