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.)