z:\doc\web\2003\11\cmprev.txt
From: Betty Peters [mailto:bettyp##ala.net]
Sent: Saturday, November 08, 2003 6:27 PM
To: Undisclosed-Recipient:;
Subject: Evaluation of CMP, one of the math book series AL will vote on
Thursday--THIS IS ONE SUGGESTED FOR MIDDLE SCHOOL FOR THE TEAM-MATH PROGRAM
IN EAST ALA -- VERY LONG ANALYSIS
California dropped these math textbooks as have quite a few other states.
Over 200 mathematicians across the nation in 2003 wrote then-Sec. of
Education Richard Riley to complain about them. The Wall St. Journal bashed
them in June 2000.
These books, and another series called "Investigations" for k-6 have been
labelled "whole math," "fuzzy math" and some in AL are calling this
"transformational math." When some gentlemen recently saw two of the
Connected Math workbooks entitled, "What Do you Expect?: Probability and
Expected Value" and "How Likely is It?" (both used in middle school grades),
they asked whether the Ala. schools were getting the students prepared for
careers in gambling! One workbook's cover had a roulette wheel and a Queen
of Hearts card; the other included a quarter and three dice. Maybe these
men are on to something because a few years ago, Mississippi authorized
courses in gambling at one its state run colleges to prepare students for
casino jobs.
An Evaluation of CMP
R. James Milgram
This report considers the National Science Foundation sponsored middle
school mathematics program, CMP, published by Dale Seymour Publishers, and
developed by G. Lappan and others, primarily at Michigan State University.
If one visits the web site of the program,
http://www.math.msu.edu/cmp/Index.html, one finds two preprints, presumably
using rigorous methodology and statistical analysis, that are advertised as
showing the benefits of CMP. Unfortunately, as we see in the appendix to
this report, both studies are fatally flawed and deceptively presented.
Additionally, at the website one will find a strong endorsement of the
program by the AAAC. They grade it as one of the most effective programs for
teaching middle school matematics Unfortunatly, this too must be taken with
a grain of salt, as is also discussed in the appendix. In fact, it is
generally acknowledged that there are no reputable studies showing that any
of the NSF developed mathematics programs actually benefit students in
testable ways.
Leaving aside these issues, we turn to the program itself.
Connected Mathematics Project consists of eight reasonably short booklets
for each of grades six, seven, and eight. The booklets for grade six are as
follows:
1) Prime Time -- factors and multiples
2) Data About Us -- statistics
3) Shapes and Designs -- two-dimensional geometry
4) Bits and Pieces I -- understanding rational numbers
5) Covering and Surrounding -- two-dimensional measurement
6) How Likely is It? -- probability
7) Bits and Pieces II -- using rational numbers
8) Ruins of Montarek -- spatial visualization
The booklets for grade seven are:
1) Variables and Patterns -- introducing algebra
2) Stretching and Shrinking -- similarity
3) Comparing and Scaling -- ration, proportion, and percent
4) Accentuate the Negative -- integers
5) Moving Straight Ahead -- linear relationships
6) Filling and Wrapping -- three-dimensional measurement
7) What Do You Expect? -- probability and expected value
8) Data Around Us -- number sense
The booklets for grade eight are:
1) Thinking with Mathematical Models -- representing relationships
2) Looking for Pythagoras -- the Pythagorean theorem
3) Growing, Growing, Growing -- exponential relationships
4) Frogs, Fleas, and Painted Cubes -- quadratics relationships
5) Say It with Symbols -- algebraic reasoning
6) Kaleidoscopes, Hubcaps, and Mirrors -- symmetry and transformations
7) Samples and Populations -- data and statistics
8) Clever Counting -- combinatorics
Overall conclusions
Overall, the program seems to be very incomplete, and I would judge that it
is aimed at underachieving students rather than normal or higher achieving
students. In itself this is not a problem unless, as is the case, the
program is advertised as being designed for all students. In fact, as
indicated, there is no reputable research at all which supports this.
The philosophy used throughout the program is that the students should
entirely construct their own knowledge and that calculators are to always be
available for calculation. This means that
* standard algorithms are never introduced, not even for adding,
subtracting, multiplying and dividing fractions
* precise definitions are never given
* repetitive practice for developing skills, such as basic
manipulative skills is never given. Consequently, in the seventh and eighth
grade booklets on algebra, there is no development of the standard skills
needed to solve linear equations, no practice with simplifying polynomials
or quotients of polynomials, no discussion of things as basic as the
standard exponent rules
* throughout the booklets, topics are introduced, usually in a single
problem and almost always indirectly -- topics which, in traditional texts
are basic and will have an entire chapter devoted to them -- and then are
dropped, never to be mentioned again. (Examples will be given throughout the
detailed analysis which follows.)
* in the booklets on probability and data analysis a huge amount of
time is spent learning rather esoteric methods for representing data, such
as stem and leaf plots, and very little attention is paid to topics like the
use and misuse of statistics. Statistics, in and of itself, is not that
important in terms of mathematical development. The main reason it is in the
curriculum is to provide students with the means to understand common uses
of statistics and to be able to understand when statistical arguments are
being used correctly.
The first four bulleted items above, particularly the second and third,
indicate areas where the program does not do an adequate job of developing
basic skills necessary for students to continue with more advanced work in
mathematics, leading to possible careers in technical areas. But even the
first cannot be ignored. It is true that the standard algorithms are not the
only methods for teaching standard computational skills, but, the skills
associated with these algorithms -- see the reviewer's discussion of
long-division, for example
http://www.csun.edu/~hcbio027/standards/conference.html/may21/milgram.html
-- as well as some training in proving algorithms correct must be developed
within the program if one is to accept the idea that students will strictly
construct their own methods. CMP simply does not do this.
Also, as noted -- while most of the topics to which the fourth bullet is
applicable are not essential for people who will never use mathematics
seriously in their professions -- for students intending careers where
mathematics is heavily used these topics can be essential.
In the detailed analysis which follows we will study three aspects of the
program. First we will look at most of the booklets for the sixth grade.
Then we will follow one important subject, exponents and exponentials, which
is primarily concentrated in the eighth grade material, and finally we will
make some brief remarks about how the program handles graphing.
The sixth grade texts:
We begin our analysis of the program with the sixth grade texts. As we go
through these booklets and a few of the more advanced ones, I will
constantly be pointing out areas where the problems above occur. This is to
help make the point that these are not isolated instances, but represent a
consistent point of view towards the material, and the level at which it
should be addressed. In fact, except for a very few instances, I do not try
to locate and point out outright errors -- though there are a number --
since errors are inevitable in the first versions of any program, and what
concerns us here are the teaching methods and objectives.
The first of the sixth grade booklets is Prime Time. This booklet is
concerned with prime factorization of whole numbers. In standards based
curricula, such as that in California, this is a fourth and fifth grade
topic (California Mathematics Standards, Grade 4, Number Sense, 4.1, 4.2,
and Grade 5, Number Sense, 1.3, 1.4), but since I view the program as
largely remedial, this is not to be regarded as a criticism.
Prime Time begins by assigning a unit project to be handed in or reported on
at the end of the unit. This project is worth noting -- here it is.
My Special Number
Many people have a number they find interesting. Choose a whole
number between 10 and 100 that you especially like.
In your journal
* record your number
* explain why you chose that number
* list three or four mathematical things about your number
* list three or four connections you can make between your number
and your world.
As you work through the investigations in Prime Time, you will learn
lots of things about numbers. Think about how these new ideas apply to your
special number, and add any new information about your number to your
journal. You may want to designate one or two "special number" pages in your
journal, where you can record this information. At the end of the unit, your
teacher will ask you to find an interesting way to report to the class about
your special number.
From both a mathematical and pedagogical point of view this is unfortunate.
Mathematically, no integers except perhaps 0, 1, and -1 are more significant
than any others. And pedagogically, this reflects a poor point of view
towards the development of the number system. If one prefers one whole
number over another, think what a big door this opens for hating complicated
fractions and even worse, irrational numbers. Basically, such a project
appears to me to be totally unjustified except in remedial situations.
In fact, this is doubly unfortunate, since, -- with exceptions that will be
noted below, but which are more or less typical of books at this level today
-- the overall discussion of numbers and their factorizations in this
booklet is first rate. The authors have access to people who know a great
deal about the subject and it shows here.
The first section in Prime Time is entitled "The Factor Game." This and the
second section "The Product Game", are about as good an introduction to
factoring whole numbers as I've seen. As their names imply these are games
that the students play with each other where winning or losing depends on
the structure of the factors in the numbers one starts with. However,
already in the second section we see a problem. Venn diagrams are introduced
towards the end of section two. But they are explicitly limited only to the
set of factors of two numbers, with the intersection region labeled by the
factors common to both. It seems to us that this is simply too limiting.
Their introduction in this way and at this point is fine. However, it is
hard to understand why there is absolutely no indication or exercise showing
that they occur in contexts other than common divisors.
This tendency of the authors to introduce important concepts and then leave
them only as tantalizing fragments will become more and more common
throughout the remaining booklets. This might be acceptable if, at least, in
the teachers manual further details were given or indications of where to
find more information, but this does not seem to be the case.
The third section, "Factor Pairs", which uses area as an interpretation of
factoring a whole number into two parts, is not quite as strong as the first
two. For example, after looking at the rectangles such as 3 by 4 and 2 by 6
obtained by factoring 12, it might be natural to draw the conclusion that
any time that a whole number is factored into a product of two whole numbers
one can draw a rectangle with the whole number as area. It would even be
natural to observe that the perimeters of such rectangles will generally not
be the same. After all, these are both fourth grade standards in California
(Grade Four, Measurement and Geometry 1.1 and 1.2). But no inferences
whatsoever are explicitly drawn, either in the teachers manual or the
student manual.
At the end of section 4, on page 43, particularly problems 19 and 20, an
excellent explanation of the sums n2 = 1 + 3 + 5 + ... + (2n+1), and 2(1 + 2
+ ... + n) = n(n + 1) is given. But once more, the general result is never
stated. In the teacher's manual, however, it is sort of stated, but not in a
way that will help an inexperienced teacher to assure that the students do
not miss the point. To make this clear, here are the comments in the
teachers manual for these problems:
19b. 1 + 3 +5+. .. + 39 = 400.
19c. row 24: 47; The sum will be 576 in row 24,
because 576 = 242. The last number in this row is 47 because 47 is
the twenty-fourth odd number. This famous pattern is the sum of the
consecutive odd numbers: the sum in each row is the square of the number of
numbers in the row.
20a. Tiles can be used to set up a visual display of this problem
(see below left). From the pattern, you can see that adding the first n
consecutive even numbers is the same as multiplying n times(n+1). So, the
next four rows are as follows:
2+4+6+8+10=30
(which is 5 x 6)
2+4+6+8+10+12 = 42
(which is 6 x 7)
2+4+6+8+10+12+14 = 56 (which is 7 x 8)
2+4+6+8+10+12+14+16 = 72 (which is 8 x 9)
20b. 2+4+6+...+40 = 420 (which is 20 x 21)
20c. row 10; 20, because 20 is the tenth even number
The fifth section on factorization leads to "discovering" the fundamental
theorem of arithmetic -- the unique decomposition of whole numbers into
products of primes. But, as usual, the theorem is never stated in the
student edition. This is particularly relevant because, though it is
possible for the students to understand what this theorem means, at this
stage it is impossible for them to have, in any way, shape or form, proved
it.
If students get the idea, based on the explorations they've made of the
meaning of the fundamental theorem of arithmetic, that they can then use it
without having been TOLD that it is, in fact, true in all cases, (and that
if they stick with mathematics long enough, they'll be given -- or construct
-- a proof), then they will have learned something VERY VERY DANGEROUS.
All too often we see students at the most advanced levels use results that
are only partially true as though they were true in every case, and serious
problems can and do result from this. But, as indicated, this is the
approach taken throughout the three year CMP sequence. I was never able to
find a place where students were warned that something which appeared to be
true after a large number of trials might fail after even more trials.
Likewise, I was unable to find any point in the program where any statement
that had been verified by the students for a number of cases was proved true
in all cases. We will discuss this further when we discuss the programs
treatment of algorithms in the seventh booklet,
Addendum: Recently a colleague who's son is currently in sixth grade in a
school system that uses CMP exclusively pointed out a very serious
difficulty with this booklet that I had not noticed originally. The material
here is not well understood by many sixth grade teachers, and the discovery
method that is used, never stating what the objective of each lesson is,
applies to the teachers manual as well. The material is not explained there
and the objectives are not stated there. I di not take this into account
when reading the teachers manuals since both of these are clear to someone
who knows the material very well.
That was not the case in this class in the Palo Alto school system: the
teacher seems to have had only the fuzziest idea of what the real objectives
of the material were, so the students dutifully did the exercises without
any guidance and developed no insight into what was happening. The result
was that the students were totally unable, by the end of the book to make
any sense at all of the locker problem.
It is critical, and even understood in a kind of general way by most
mathematics educators, that teachers must understand the material even
better in a discovery situation than they need to when the instruction and
material are more traditional.
Bits and Pieces
It would also have been natural at this point to introduce exponents --
which is a fifth grade standard in California (Grade 5, Number Sense, 1.3)
-- but this is not done. In fact the first mention that I was able to find
of exponential notation occurs on page 42 of the final seventh grade
booklet, Data Around Us.
Finally, the sixth section, the locker problem is excellent. However, here,
relating the evenness or oddness of the number of factors of a number to the
result in the end absolutely begs for further examples, such as, e.g., the
Konigsberg bridge problem. The point is that the process of understanding
why only certain lockers are left open after a number of students have
passed through is a pure process of abstraction. The method of thought
implicit here is the same method used to solve the Konigsberg bridge
problem, though the contexts initially appear to be totally different. This
is another of these examples where something good is started but left
hanging.
Overall, though, this is a good set of lessons. In a curriculum such as that
in California it could be used to good effect as a supplement for the normal
fifth grade material, and to help with remediation in the sixth grade. In
these contexts the failings noted above would not be significant. But Prime
Time is also the high point of the five sixth grade booklets that I have
examined. The others range from significantly less good to very bad indeed,
as the problems noted above become more and more significant.
The next booklet that we shall examine is"Data About Us".
The first section works well as an introduction to the subject of data
collection.
The second section, "Types of Data""starts out badly however. The second
paragraph reads:
When we collect data, we are collecting a measurement" about some
"thing." We are interested in organizing the data by tallying, or finding
the frequency of occurrence for each data value."
Of course, data is data. We introduce measurement as a means of describing
data.
Except for this, the material is sound, and, overall the material is well
covered. However, one should be aware that this material, in California at
least, and certainly in countries like Japan and Singapore is covered one to
two years earlier than sixth grade. (California Math Standards, Grade 4,
Statistics, Data Analysis, and Probability, 1.1, 1.2, 1.3 and Grade 5, 1.1,
1.2, and 1.3) In fact the standards mentioned go significantly further than
the material covered in Data About Us. A characteristic of the discussion
throughout, is a superficial analysis of data and simply a description of
the meanings of basic terms, but no precise definitions. In the student
material for the fifth section, "What do we mean by the mean, the term mean
is NEVER defined explicitly.
In summary, this booklet has a distinctly remedial character and, while the
material is well organized and important, it is treated purely
descriptively, not precisely. Terms, even basic terms like mean, are never
defined.
Here, even more so than with Prime Time, one can imagine that the best use
of this material is in a situation where the students are distinctly behind
their expected level and have not had any experience of the precision of
mathematics. In fact, in the local district where I live, both Data About Us
and Prime Time were used in exactly this kind of situation -- a sixth grade
class where the students had used Mathland in their earlier grades and had,
consequently, extremely weak basic skills. In this situation the CMP
booklets were very well received by the students, and the teachers reported
that the students skill levels improved dramatically.
Addendum: In the case of the Palo Alto sixth grade class, the results with
this text were also very disappointing. The lacks mentioned above were
magnified by the lack of understanding of the material by the teacher and
the failure of the teachers manual to offer any detailed help. Once more it
is reported that the result was complete confusion on the part of the the
students. By contrast, in the class in my local district, it was evident
that the teacher knew the material throughly, commenting repeatedly when
discussing both this booklet and the previous one with me, that she had
known the material in a general way previously, but that seeing it developed
in this way, and having to reconstruct it for herself, clarified it for her
enormously. We can infer from this that it was the process of understanding
going on with this teacher which enabled her to be successful in using the
booklet with her class.
It appears essential to reiterate my previous observation that in a
discovery situation it is absolutely necessary that the teacher understand
the material considerably better than is required in a more traditional
environment. It is likewise necessary to note that it is totally unrealistic
to expect this from the vast majority of the teachers in this nations middle
schools.
The next book in the series is Shapes and Designs. Here the remedial nature
of the program becomes even clearer. What follows is the description of the
individual sections from the teachers manual.
Investigation 1: Bees and Polygons
This investigation poses the key question, What tile shapes can be
used to cover the plane? It asks students to make conjectures about why
honeycombs are covered with hexagons and to use physical materials to
explore other possibilities.
Investigation 2: Building Polygons
This investigation is based on the general question, Is the shape of
a polygon determined exactly by the lengths of its sides and the order in
which those sides are connected? The three problems involve the use of
manipulatives called Polystrips.
Investigation 3: Polygons and Angles
This investigation introduces three basic ways of thinking about
angles and the ideas behind angle measurement. It gives students practice in
estimating angle measurements based on a right angle. A measuring device,
the angle ruler is introduced, allowing more precise measures of angles.
Students then explore a problem that looks at the possible consequences of
making measurement errors.
Investigation 4: Polygon Properties and Tiling
This investigation focuses attention on some basic properties of
familiar quadrilaterals, using tiling as a context.
Investigation 5; Side-Angle-Shape Connections
In this investigation, students look at what remains constant and
what changes as triangles, squares, rectangles, and parallelograms are
rotated and flipped. The symmetries of the figures become more evident as
students work with them.
Investigation 6; Turtle Tracks
In this investigation, students use the logo programming language to
create computer designs, two of the three problems in this investigation can
be done even if students do not have access to computers.
As an example of level, compare the California standards (Grade four,
Measurement and Geometry, 3.3, 3.4, 3.5, and 3.6 as well as the fifth grade
Measurement and Geometry standards, 2.1, 2.2, and 2.3). Indeed, in checking
the further geometry booklets in grades seven and eight of the CMP program,
we find that in total, they cover no more than the material spelled out in
these fourth and fifth grade standards.
The discussion in Shapes and Design ultimately focuses on angle measure but
never really states anything specifically. A great deal of time is spent
MEASURING angles with a protractor, to the point where several problems are
given on page 38 listing series of measurements of the same angle and asking
questions like ""What method would you use to decide on the best measurement
for each angle?" This does not seem to be well designed as preparation for
geometry where the focus is on the abstract properties of precisely known
angles and lengths. Moreover, though it is assumed in the teachers' manual
that the teacher knows that the sum of the interior angles of a triangle is
180 degrees, I searched in vain for any explicit statement of this in the
student material. The students are, presumably, supposed to figure this out
for themselves with their inaccurate angle measurements.
It is precisely at this point that CMP becomes strictly remedial. If
students are to go on to higher levels of achievement in mathematics, from
geometry through calculus, linear algebra and beyond, they must be able to
handle precisely defined abstract concepts. Moreover, these abilities are
difficult for even the strongest students to master, and they take
considerable time to develop.
Pages 40 and 41 in Shapes and Designs are very bad with respect to the
considerations above. Here definitions are confused with measurements in the
case of the angles that a transverse line makes with "parallel" lines. (The
point is that "parallel" is really an abstract concept, and we cannot decide
if two lines are parallel by "real" measurements which always have some
errors.)
Incidentally, at this point, all the authors had to do was just mention as a
fact that the angles of intersection of transversals with parallel lines are
the same, and all the material needed to demonstrate that the sum of the
interior angles of the triangle is 180 degrees would have been available.
But as I've indicated is typical in this program, they promptly leave the
subject hanging on page 40 and don't seem to return to it again in this
booklet. However, on page 50 of Shapes and Designs we find the following:
These questions will help you summarize what you have learned:
a. In regular polygons, what patterns relate the number of sides to
the angle sum and the size of the interior angles?
b. In irregular polygons, what patterns relate the number of sides
to the angle sam and the size of the interior angles?
Think about your answers to these questions, discuss your ideas with
other students and your teacher, and then write a summary of your findings
in your journal.
The final section, "Turtle Tracks", uses turtle graphics through logo to
give students some experience with computer programming. By comparison, very
similar problems occur in the third and fourth grade texts of the
McGraw-Hill SRA series,"Explorations.
Here is another problem I had with this booklet. On page 21c there is the
following sidebar:
For the Teacher: Generalizing Mathematical Statements
Some teachers take this opportunity to discuss with students how
mathematicians think and how they record the results of their
experimentation:
"This is not something you are responsible for knowing, but I want
to
show you how mathematicians would use the language of mathematics
to record your generalization. Mathematicians try to talk about ideas at
a general level, rather than about a specific case. For example, they give
names to the lengths of a triangle's sides rather than talking about a
triangle with specific sides like 8 cm, 5 cm, and 6 cm. Instead, they call
the sides of a triangle by letters, such as side a, side b, and side c. So a
triangle with sides a, b, and c stands for any triangle you can make.
Mathematicians would write your statement like this:"
If a and b represent the two shorter sides of a triangle and c represents
the longest side, then a + b > c.
In one sense, this is the beginning of something potentially very important
-- an effective introduction of the process of abstraction. But note how it
is introduced: students could be made aware that "mathematicians" think
about things in this way. There is no indication that students could benefit
by trying to think in this way. In another sense this is not by any means an
accurate description of either mathematicians or the process of doing
mathematics. Things are stated in generality only when the statement is
sufficiently important and useful that a general statement is merited. There
are innumerable papers in the mathematical literature giving detailed
analyses of single examples.
We now turn to the seventh booklet, Bits and Pieces II, which is the second
booklet discussing rational numbers. Here are the overview and the the
author's description of the mathematics in this booklet.
Rational numbers are the heart of the middle-grades experiences with
number concepts. From
classroom experience, we know that the concepts of fractions, decimals, and
percents can be
difficult for students. From research on student learning, we know that part
of the reason for
students' confusion about rational numbers is a result of the rush to symbol
manipulation with
fractions and decimals.
In Bits and Pieces I, the first unit on rational numbers, the
investigations asked students to
make sense of the meaning of fractions, decimals, and percents In different
contexts. In Bits and
Pieces II, students will use these new numbers to help make sense of many
different situations.
The Mathematics in Bits and Pieces II
This unit does not teach specific algorithms for working with
rational numbers. Instead, it helps
the teacher create a classroom environment where students consider
interesting problems in
which ideas of fractions, decimals, and percents are embedded. Students bump
into these impor-
tant ideas as they struggle to make sense of problem situations. As they
work individually, in
groups, and as a whole class on the problems, they will find ways of
thinking about and operat-
ing with rational numbers.
The teacher's role is to help students make explicit their growing
ideas about the world of ratio-
nal numbers and, when students are ready, to inject ideas and strategies
into the conversation
along with the ideas and strategies generated by the students. Simply giving
students algorithms
for moving symbols for rational numbers around on paper would be a mistake
and the tempta-
tion to do so is often great. All teachers want their students to succeed,
and showing them how
to do something such as how to cross multiply to compare two fractions gives
the impres-
sion of immediate success. Students can do the algorithm by memorizing.
However, evidence
from student assessments shows that students do not understand algorithms
that are given to
them in this way and therefore cannot remember or figure out what to do in a
given situation.
This unit provides a rich set of experiences that focus on
developing meaning for computations
with rational numbers. We expect students to finish this unit knowing
algorithms for computa-
tion that they understand and can use with facility.
The discussion above of the mathematics in Bits and Pieces II represents a
highly controversial point of view about the subject. This view is agreed
with by less than 1% of the professional mathematicians in California, for
example. No one would dispute the argument that rote memorization of
algorithms alone does not lead to understanding. However, when an algorithm
is introduced together with a careful and precise explanation of how and why
it works, students are exposed to material that is critical to the continued
development of their mathematical skills. Whether students learn these types
of things using discovery methods or other methods is not important. What is
critical is that they learn them somehow.
Now we turn to the individual sections of Bits and Pieces II.
The first section discusses percents, and concentrates on percent reductions
in prices, sales taxes, and tips. It is grade appropriate and solid
mathematically. It is, in fact, among the best presentations of this topic
at the sixth grade level that I've seen in the sense that the ideas are
clearly explained and evidently understood by the authors. However, the
material here is not really theory. These are concepts that play a major
role in everyday life. It is a start (and an important one) that the
concepts be clearly enunciated. But these topics demand mastery level
learning on the part of the students.
At this point the discovery method and nothing else philosophy of the
authors definitely works to the detriment of the students. The numerical
skills involved with these topics require practice on the students' part,
and discovery methods do not encourage this. Indeed, the discovery approach
is carried to extreme levels here. For example note the quote on page 164 of
the teachers manual in a sample letter meant to be sent to parents: "It is
important that you do not show your child rules or formulas for working with
fractions. This unit helps students to discover these rules for themselves .
. . ." So, in spite of the basically solid exposition, if the subject is
taught as the authors seem to intend, there is every reason to expect that
the students will not learn the material to the depth required.
The continuation in the second section maintains the high level of
exposition found in the first. Here percents are also tied in to topics in
the data analysis and statistics strand. The reservations indicated above
are less compelling here as the material is not quite as basic.
The third section is concerned with estimating using fractions and decimals.
The discussion is developed through the notion of deciding when a benchmark
fraction represented on the number line is closest to another number. But in
the first example, they already show a difficulty with this by using an
example where the number is exactly half-way between the two closest
benchmark numbers. As usual, however, they leave this hanging. Then these
benchmark numbers are used to estimate sums of fractions and a game,
"Getting Close" is introduced. A large number of problems involved with
various aspects of estimation are then given, both numeric and geometric.
My personal view is that too much time is devoted to estimation. The costs
here are in diluting the precision of the abstract concept of a fraction
with the approximate nature of many "real world" applications. One of the
chief aims of a traditional education in mathematics was to give students
experience with precise thinking -- the ultimate aim being to aid them in
making reasoned, rational decisions -- and the approach to manipulating
fractions here is not well aligned with that objective. But this is a matter
of opinion and should not exactly be taken as an objection.
The third section is well done, taking its objectives into account. But it
is important to realize that my view of these objectives is that they are
aimed at weaker students. Diluting the precision of the concept of a
fraction cannot possibly be of help to students intending to go on to study
advanced topics in mathematics such as those required for careers in
engineering, economics, or other related technical areas. Here students must
be prepared to deal on an everyday basis with things like Laplace transforms
which convert systems of linear differential equations into matrices whose
entries are quotients of polynomials. If the concept of a fraction is not
crystal clear to these students, they will have severe difficulties at this
point. Indeed, too often in recent years, this is exactly what I've seen
even in classes at Stanford.
The fourth section is a different story. Here the authors mix approximation
with exact numbers, totally confusing the two through the means of a land
map, which describes, using straight lines but no indications of length or
area, the decomposition of two sections of land among many owners. Then it
is assumed that various sales took place and precise contiguous areas e.g.,
1/2 of one section are supposed to have ended up in the hands of only four
of the original owners. This mix of precision and imprecision is never
clarified and is used as the basis for explaining addition and subtraction
of fractions.
At this point, consistent with the point of view towards algorithms
described in the introduction to this booklet, the students are asked to
work in groups and find their own algorithms for adding and subtracting
fractions. Moreover, as verifications of correctness they are given the
following instructions:
Test your algorithms on a few problems.... If necessary, make
adjustments to you algorithms until you think that will work all the time.
Write up a final version of each algorithm. Make sure they are neat and
precise so others can follow them."
The simplest method for achieving this is to simply make a list of the test
problems and their answers, and the algorithm would be -- locate the set
problem on the list and write down the answer.
But to add insult to the anti-mathematics above, on page 48, the same page
as the directions above, the following definition of an algorithm is given:
To become skillful at handling situations that call for the addition
and subtraction of fractions, you need a good plan for carrying out your
computations. In mathematics, a plan -- or a series of steps -- for doing a
computation is called an algorithm. For an algorithm to be useful, each step
should be clear and precise so that other people will be able to carry out
the steps and get correct answers.
This is incorrect. What they have defined is a "program," and even this is
not quite right. A central issue in the development of both programs and
algorithms is the issue of correctness. This is subtle but crucial. If
students are going to develop programs and algorithms but are never shown
how to prove them correct, disasters occur. On the other hand, at this level
it may be very difficult to demonstrate the correctness -- or more likely,
incorrectness -- of a student provided algorithm.
Approximately 1 in 10 of the students I've had in recent years in the
differential equations course at Stanford have believed that (a/b) + (c/d) =
(a+c)/(b+d). It is somewhat difficult to be comfortable with an engineer who
does his/her calculations in this manner.
In the fifth section, to illustrate that the material above was no accident,
it is proposed, on page 59 that the students work in groups to develop their
own algorithms for multiplying fractions. The test of correctness is a
repeat of the directions above.
The final section, "Computing with Decimals" is better, assuming that the
students have survived sections 4 and 5 and actually have correct methods
for adding, subtracting, multiplying and, as they say in the introduction
"possibly dividing decimals." The explanations and exercises here seem to
actually be helpful.
But as is becoming more and more the norm with this program, there is a
difficulty. Here is the main part of page 69:
When you multiply 0.1 by 0.1 on your calculator, you get 0.01. What
is the fraction name for 0.01? It is 1/100, as you saw with the grid model.
In the next problem, you explore what happens when you multiply
decimals on your calculator. Before you use a calculator to find an exact
answer, think about how big you expect the answer to be..
Problem 6.3
A. Look at each set of multiplication problems below. Estimate how
large you expect the answer to each problem to be. Will the answer be larger
or smaller than 1? Will it be larger or smaller that 1/2?
Set 1
21x1 =
21 x 0.1 = etc.
21 x 0.01 =
21 x 0.001 =
21 x 0.0001 =
B. Use your calculator to do the multiplication, and record the
answers in an organized way so that you can look for patterns. Describe any
patterns that you see.
Now we see why the authors believe that they can proceed in the way
indicated. The students do not, in fact, have to learn to actually add,
subtract, multiply, or above all, divide decimals, since their calculators
will do it for them.
Unfortunately, as has been shown in work on curricular development, many of
the cognate skills implicit in things like learning the long division
algorithm become important in different contexts many years later.
Consequently, students who have not developed these skills often seem to
find themselves at a serious disadvantage when attempting to work in
technical fields.
All in all, Bits and Pieces II, in spite of the initially good exposition in
the first two sections and part of the last, is a very poor booklet, and
probably does more harm than good.
The last booklet in the series for sixth grade is Ruins of Montarek. Here is
the last paragraph of the overview in the teachers manual. It is clear from
this paragraph that the material here is entirely remedial, and this is
consistent with the content which correspond with the California Third Grade
Standards (Measurement and Geometry, 1.1, 1.2)
Spatial visualization skills are very important in developing
mathematical thinking and are critical to reading graphical information,
using arrays and networks, and understanding the fundamental ideas of
calculus. In the past several decades, research has raised many questions
about spatial visualization abilities. Many studies have found that girls do
not reason as well about spatial experiences as do boys, especially starting
at about adolescence. The explanation offered by some psychologists, that
this difference may be innate, is unacceptable to those of us concerned with
teaching children.
It might also be worth noting that the role of spatial visualization in
topics such as calculus is overstated. In fact the vast majority of the
topic takes place in two dimensions. When one finally gets to questions in
three dimensions involving solid integrals, surface area, and related
topics, virtually all students appear to have serious difficulties, and the
"skills" developed in this booklet are not going to address the problem
areas which actually occur.
This completes our review of the sixth grade material in this program.
Throughout, our perspective has been to illustrate the ways in which it
differs in essential ways from more standard programs. We now turn to a
discussion of some of the eighth grade material to illustrate the fact that
these differences remain consistent throughout the entire program.
The handling of exponents and exponentials in CMP
In traditional texts, the Japanese texts, and most others, the order in
which exponentials is done is to first introduce exponential notation,
explain the exponent rules in the case where the exponents are positive
integers, explain that a0 = 1, and using this, explain that a-n = 1/an.
After this students learn about fractional exponents. (Of course it helps
enormously if the students have already discussed topics like square roots
and maybe cube roots.) Finally, if there is time, the general form ax is
introduced for any number x (and positive a). The cognate topics here that
it is natural and perhaps necessary to discuss are things like the existence
of irrational numbers and the fact that numbers like the square root of 2
are irrational.
Also, traditionally, one of the main methods of introducing exponents was
through compound interest, a standard seventh grade topic.
In short, developing a reasonably full discussion of exponential
relationships involves a major effort.
The booklet Growing, Growing, Growing is the only one among the 24 booklets
to discuss any of these topics with the exception of Frogs, Fleas, and
Painted Cubes, which discusses quadratics but is recommended for discussion
after Growing, Growing, Growing, and, in Thinking with Mathematical Models
which precedes Growing, Growing, Growing, a discussion of functions of the
form a + bx-1, (though written in the form a + b/x), and a single example of
compound interest (again done without using the exponential notation).
Consequently, before we discuss Growing, Growing, Growing in detail we will
briefly discuss the two sections in Thinking with Mathematical Models which
involve exponentials. In section two, "non-linear models," the discussion
starts with a physics experiment using beams made of paper. The students are
to suspend the beams at their endpoints and successively pile pennies on the
center till the beams crumple. They then graph the results and, hopefully,
notice that the result is non-linear. No discussion of the physics involved
is given, nor could there be at this level. Finally, the discussion focuses
on the equations above, and the general shape of the graphs are described.
The third section, "More Nonlinear Models" in Thinking with Mathematical
Models, starts with an elementary example of compound interest. Then it
turns to another experiment -- this time with a glass filled with water.
Half the water in the first glass is poured into a second glass, half the
water in the second glass is poured into a third, and so on. The students
are supposed to then notice the inverse exponential shape of the resulting
water levels. The remainder of the discussion here takes place in the
problems.
Both of these sections are entirely descriptive. The final problem in the
third section will give an idea of the depth of the discussion.
7. Four biologists are studying the caribou and wolf populations in a
particular area of Alaska. The caribou are prey to the wolves, and keeping
the two populations in balance is important to the survival of both species.
The biologists are trying to predict what will happen if no measures are
taken to control the populations. After studying the situation, each
biologist makes a graph of his or her prediction. Describe what each graph
represents in terms of the animal populations.
The four graphs are respectively a straight line parallel to the x-axis, a
mildly concave (downward) curve, a sharply concave curve but with the base
labeled wolves and a straight line with negative slope. In all four cases
the y-axes is labeled caribou, and in the remaining three cases the x-axis
is labeled time.
The actual mathematical issues involved -- for example the standard
non-linear differential equations modeling predator-prey situations -- are
far beyond the level of the students. Also, the answers given are not
realistic. In point of fact, what tends to happen is that as the caribou
population declines, the wolf population initially rises but then also
declines which allows the caribou population to increase, and the situation
cycles. (This is the situation near the stable point of the Volterra
predator-prey equations. ) The situation of the first graph occurs only at
the single fixed point, so should NEVER be observed in nature.
Now let us turn to the booklet Growing, Growing, Growing. In a standard
algebra course, one of the key topics is exponents. In fact, usually,
exponents have been introduced in seventh grade, and maybe even sixth grade
with squares and cubes being written in the exponent notation. Also, in
seventh grade it is not unusual that some fractional exponents have been
introduced -- at the least, square roots. In any case, in a standard course
students are expected to learn and understand the exponent laws
a(m + n) = a m a n and amn = (a m) n. Here is the totality of the discussion
of the exponent laws in Growing, Growing, Growing.
2. Cesar said that since he can group 2x2x2x2x2x2x2x2x2x2
as(2x2x2x2) x (2x2x2x2x2x2), it must be true that 210 = 24 x 26
a. Verify that Cesar is correct by evaluating both sides of the
equation
210 = 24 x 26.
b. Use Cesar's idea of grouping factors to write three other
expressions that are equivalent to 210. Evaluate each expression you find to
verify that it is equivalent to 210
c. The standard form for 27 is 128, and the standard form for 25 is
32. Use
these facts to evaluate 212. Show your work.
d. Test Cesar's idea to see if it works for exponential expressions
with other bases, such as 38 or 1.511 lfest several cases. Give an argument
supporting your conclusion.
e. Find a general way to express Cesar's idea in words and with
symbols.
Extensions_______
13. Molly figured out that 26 = 64 and 43 = 64. Then, since 22 = 4,
she substituted 22 for 4 in the expression 43 and got (22)3 = 64. She said
that since 26 = 64 and
(22)3 = 64, it must be true that (22)3 = 26.
a. Verify that Molly is correct by evaluating both sides of the
equation
(22)3 = 26.
b. Use Molly's idea to find an exponential expression equivalent to
the given expression.
i. (34)2
ii. (43)2
c. Find a general way to express Molly's idea in words and with
symbols. Check your idea by testing it on three more examples.
In this instance, as is typical of the program, an extremely important topic
is introduced and immediately dropped. Here is the next problem.
14. Juan wrote out the first 12 powers of 2. He wrote 21 = 2,
22 = 4, 23 = 8, and so on. He noticed a pattern in the digits in the
units places of the results. He said he could use this pattern to predict
the digit in the units place of 2100.
a. What pattern did Juan observe?
b. What digit is in the units place of 2100. Explain how you found
your answer.
Of course, what is going on here is arithmetic modulo 10. This is a much
more sophisticated topic, and one seldom felt necessary to discuss in K -
12, since it has very limited applicability. Moreover, as is evident, no
attempt is made to explain what is going on, and certainly the students are
not asked to justify or prove their answers. They are expected, - as the
"solution" in the teacher's guide shows - to notice after eight trials that
there appears to be a repeating pattern 2, 4, 8, 6, 2, 4, 8, 6, and to guess
that this is, in fact, the general case. Mathematically speaking this is a
disaster, and would be unacceptable in a program designed for students
requiring serious mathematical backgrounds.
The next section, "Growth Patterns" is again purely descriptive. The basic
intent is to introduce the concept that exponential growth is characterized
by the property that the value at stage n is a constant times the value at
stage (n-1). The material here serves a useful purpose as an introduction to
exponential growth and decay. However, the actual mathematics involved in
any kind of serious study of these topics is quite a bit more advanced,
almost inevitably requiring calculus. Consequently, it is traditionally
deferred to a much later stage in the curriculum. For example it is not
discussed before grade 10 in the Japanese books discussed above. Indeed, the
following problem in this second section illustrates the level of the
discussion here:
11. Calculators use scientific notation to display very large results.
For each expression, find the largest whole-number value of n for which your
calculator will display the result in standard notation.
a. 3n
b. pn
c. 12n
d. 237n
The next section "Growth Factors" considers exponential growth as before ,
the only difference being that now the multipliers are no longer required to
be whole numbers. There is a good introduction to compound interest here,
though only the growth of value of an investment is considered.
Here is one of the final exercises in this section:
17. If your calculator did not have an exponent key, you could evaluate
1.512 by entering 1.5 x 1.5 x 1.5 x 1.5 x 1.5 x 1.5 X 1.5 x 1.5 x 1.5 x 1.5
x 1.5 x 1.5.
18. How could you evaluate 1.512 with fewer keystrokes
19. What is the least number of times you could press x to evaluate
1.512?
What is unexpected here is the answer to the second part given in the
teachers manual:
Answers will vary. The calculation given in the answer to the first
part requires four presses of x, as does this calculation: 1.5 x 1.5 = 2.25,
2.25 x 2.25 x 2.25 = 11.390625, 11.390625 x 11.390625 = 129.7463379.
This is nonsense. There can only be one correct answer to any question which
asks for a least number. In fact, the minimum possible is four. On the other
hand, this is another example of a puzzle problem in the book, which, as far
as I can see, leads to no interesting or important developments in the
subject.
This concludes our discussion of the handling of exponents in CMP. Again,
let me emphasize that for the most part what I've tried to do is to point
out the distinction between the handling of this very important topic in
CMP, and what would be expected in a more standard program, geared to
developing skills needed in more advanced areas of mathematics.
The handling of graphs in CMP
Let us conclude this review by considering a crucial part of the treatment
of graphs in the eighth grade component of CMP, and comparing it to the
handling of this topic in the seventh grade Japanese texts.
In the seventh grade algebra CMP text Moving Straight Ahead, graphs of
linear equations are considered, and in a few places the intersections of
the graphs of two different linear equations are considered, but no general
methods seem to be introduced for determining the intersection. For example
there is a related rate problem involving a race between two brothers, Henri
and Emile, in 2.5. In 3.1, the booklet takes up the discussion of this
probem again with the following remarks found on page 52h.
The point of intersection is the point at which Emile overtakes
Henri. The boys will be at the same distance from the starting line and will
have walked the same amount of time.
What are the coordinates of the point of intersection? (The
intersection occurs at t=30 seconds and d = 75 meters.)
How did you find the point of intersection?
Students will have found this in Problem 2.5 by making a graph by
hand.
Explain that this point can also be found by using a graphing
calculator. Enter the equations into your overhead graphing calculator (if
you have one), and have students do the same. ...
Incidentally, there is a curious problem on page 75. Problem 29 is given as
follows:
29. In 1980, the town of Rio Rancho, located on a mesa outside Santa Fe,
New Mexico, was destined for obscurity. But as a result of hard work by its
city officials, it began adding manufacturing jobs at a fast rate. As a
result, the city's population grew 239% form 1980 to 1990, making Rio Rancho
the fastest-growing "small city" in the United States. The population of Rio
Rancho in 1990 was 37,000.
a. What was the poplulation of Rio Rancho in 1980?
b. If the same rate of population increase continues, what will the
population be in the year 2000?
The answer given for 29(a) is 2.39P = 37,000 so P = 15,481 people in 1980. I
guess we should be glad, that the population did not increase 0%. To make it
clear that this is not an accident, here is the answer for 29(b). P =
2.39(37,000) = 88,430 people in 2000. Actually, I had not initially noticed
problem 29 because of this. The thing that initially interested me was that
they had misplaced Rio Rancho by about 50 miles.
The final section of A World of Patterns, is introduced as follows: "In this
investigation you will sketch graphs that fit written descriptions, and you
will make up stories about what a given graph might represent." Remember
that this is supposed to be an eighth grade algebra text. This appears to be
a discussion of graphs that would be more appropriate in a much lower grade,
for example compare the California standard, (Grade five, Algebra and
Functions, 1.1).
To illustrate the low level of skills developed in this booklet consider
problem 7 on page 55.
7. In your previous math work, you investigated the relationships among
the radius, height, base area, and volume of a cylinder. You found that the
volume of a cylinder is equal to its base area multiplied by its height.
a. Suppose you are in charge of designing a cylindrical can to hold
250 ml of juice. Investigate some possible (base area, height) combinations
for the can. Try radii of 2.5 cm, 3 cm, 3.5 cm, and any other measurements
you think are reasonable. Record your finding in a table.
b. Make
a graph of your (base area, height) data.
c. Draw a straight line or curve to model the data. What other
situations in this unit have similar graph models?
d. Write an equation that fits your graph model.
e. Which (base area, height) combination would you choose for the
can? Give reason for your answer.
Notice that there is no requirement at all that the students do any symbolic
manipulations. In a more standard text here is what might be done. The
volume is given by the formula V = (pi)r2h, and this is assumed constant.
The students might well be asked to determine the surface area of a can with
constant volume, which would be given by the formula 2(pi)(r2 + rh), so
substituting for h,
A = 2(pi)(r2 + V/((pi)r)).
They could then graph this to estimate a the size of a can with minimal
area. (It would be asking too much to have them determine the minimum
exactly.) But this variant of the CMP problem is at a reasonable level for
an eighth grade algebra text. It is also worth noting that the basic
formulae for the area and volume suggested above are contained in the
California sixth grade standards, (Measurement and Geometry, 1.2, 1.3, and
Algebra and Functions, 3.1, 3.2).
Let us consider, by comparison, the handling of the topic of graphs in the
Japanese seventh grade text from 1984 translated and published by The
University of Chicago School Mathematics Project in 1992.
Their chapter on functions starts on page 97 in a similarly descriptive way
with the description of the height of a meteorological rocket as a function
of time. It then presents a precise definition of a function on page 99.
When quantities vary in accordance with changes in other quantities,
all these quantities are expressed as variables such as x and y. If we
determine the value of x, the value of y is also determined. In situations
like this, we say that y is a function of x.
On page 101 it introduces proportions and inverse proportions with the
following remark.
You learned about proportions and inverse proportions in elementary
school. Now we will learn about functions which are defined by proportions
and inverse proportions.
In the problems on functions and proportions on page 107 here is the second
problem:
* The bottom of a rectangular container is 40cm long and 20cm wide. If
we let 200 cm3 of water into the container every second for t seconds, the
depth of the water becomes h cm. Answer the following problems:
* Express h in terms of t and show that h is proportional to t.
* Is t proportional to h? If so, state the constant of
proportionality.
Then a discussion of coordinates and graphs is initiated on page 108. Once
more it is pointed out that graphs of linear equations had already been
learned in elementary school, but only in the first quadrant, and that in
this section they will extend their knowledge to negative numbers as well.
The section concludes with a study of functions of the form y = ax-1, and
the precise definition is given:
Generally, when a is a nonzero constant, the graph of y = ax-1
consists of two smooth curves. This curve is called a hyperbola.
The Japanese program is similar to CMP in these grades in that it is
"integrated," and programs like CMP are supposedly developed on the Japanese
model. However, as indicated above, there is a dramatic difference in level
between the two programs. In terms of content and precision of the treatment
of the various topics a traditional US program would tend to be much nearer
to the Japanese model . The main difference is in the fact that the
traditional US programs tend to cover many fewer topics each year in the
higher grades.
Graphs are also considered in a few of the other books, for example the
eighth grade booklet Thinking with Mathematical Models and the seventh grade
booklet, Variables and Patterns, but in all cases the discussion is at or
below the level indicated above.
Appendix: The supporting literature for CMP
I started my evaluation of the middle school program -- Connected
Mathematics Project -- by visiting their web-site
http://www.educ.msu.edu/cmp/, and printing out the article Effects of the
Connected Mathematics Project on Student Attainment by M.N. Hoover, J.S.
Zawojewski , and J. Ridgway. There is a brief analysis of the article by W.
Bishop that can be found at http://lynch.nscl.msu.edu/tsang/eval1.htm.
Perhaps the most interesting and important datum in the report is the
following graph:
It is accompanied by the following text:
A second issue raised by this data is the role of computation and
the different picture of computation across the three grades. For example,
at both sixth and seventh grades adding Computation to Math Total lowers the
CMP gain scores while raising the Non-CMP gain scores. Furthermore, with
Computation included, CMP gains statistically less at sixth grade, gains the
same at seventh grade, and gains statistically more at eighth grade. What
would account for these patterns? Figure 2 shows displays this data
graphically.
On the face of it, this graph seems to imply that there is a consistent
improvement in the scores of the CMP students when compared to non-CMP
students. One mathematician who has carefully read this paper had the
following observations.
Look at the graph called Figure 2 "Fall to Spring ITBS Scores". You
will notice that the non-CMP students seem to get dumber as the years go by,
and seem to forget more over the summers than they learn during the year.
The reason is obvious: the non-CMP students were 3 separate groups of
students, not the same set over three years, (On page 2, the paper suggests
that this is the case). NO effort was made to calibrate differences. It is
pretty clear what is happening: the grade 6 data is from a school where the
honors kids were non-CMP and the grade 8 data is from a school where only
remedial kids were non-CMP.
They do not even try to disguise this.
There is a second paper extolling CMP at the CMP website by Reys, et.al. In
this paper the statistics are done well but the "control group" is not
realistic. The paper looks at three programs: CMP, another similar program,
and a "control group" that consists of teachers who seem to share the same
philosophy as the developers of CMP but are teaching without the assistance
of any books or course materials. In other words the control group consists
of teachers who are just winging it.
Unfortunately, this kind of statistical analysis, poorly done and
misleading, appears to be very common in research on NSF funded programs,
and the errors all seem to be in the direction most favorable to the
programs. For example one can check Kim Mackey's analysis of similar
research reports on CorePlus on the math-teach archive at Swarthmore:
http://forum.swarthmore.edu/epigone/math-teach
April 11 - 14, 1999, Core-Plus Evaluation, Parts I - IV.
Finally, the site contains a ringing endorsement from the AAAS. Here are the
comments of one of the professors in the department of mathematics at
Michigan State explaining the significance of this endorsement.
"I am a MSU Math professor. While I am a research mathematician, I have been
teaching courses in Math Education, and have been closely following current
developments in Math education. Evaluating Math programs is a tricky
business in the current environment. There are major battles going on, and
reports from even trusted sources are usually tinged with the politics of
these battles.
The AAAS report is a case in point. It was not written by scientists.
Rather, the AAAS has lent its imprimatur, under the name `Project 2061' , to
a group of EDUCATORS who are not trained in mathematics or science. These
educators have a specific political agenda. They would like to see all
education done in group settings with the `discovery method' and with no
direct instruction from the teacher. They would like to see mathematics
classes with long writing assignments, no right and wrong answers, no
practice problems, complete reliance on calculators, and a minimization of
algebra.
Accordingly, they designed a set of criteria focusing not on WHAT and HOW
mathematics is covered by the program, but rather on the extent to which it
conforms to the above agenda. Take a close look at the Project 2061 website
http://project206 1.aaas.org/newsinfo/press/attach_a.pdf
and you will quickly see that this is so."