The Bellevue Public School (BPS) District has decided to choose and implement a new high school mathematics curriculum to replace the four years of algebra through pre-calculus. This program was developed by a group called “Core Plus” or CPMP (“Core Plus Mathematics Program”) while the title of the 3 (or 4) year course is Contemporary Mathematics in Context. By being a noisy parent, I was briefly loaned a copy of the CPMP materials. The selection of CPMP was proposed to the school board May 4, 1999 before I ever saw the materials. In fact, the BPS School Board approved adopting the program at a special (closed) session May 18 before I – or any other parent -- saw the materials. Interestingly, when the May 18 special board meeting was scheduled, its purpose was limited to the board learning more about the program from staff before deciding on the program. As math in Bellevue goes sailing down the commode, we can be comforted that at least the doors are closed and the lights are off.
If CPMP were installed as a program to salvage the bottom 15% to 25% of Bellevue students, I would support that introduction. However, CPMP is to be applied to all students regardless of interest or capability. This really is inappropriate, especially without any prior discussion with parents. Even the CPMP implementation manual advises school districts to meet with parents prior to the introduction of the courses to explain the program and let parents have a chance to see the materials. As of the day before introduction, I appear to be the only parent not on the school board who has examined any CPMP materials.
The concerns I have with CPMP appear to apply to the other mathematics programs recently adopted in Bellevue (CMP for grades 6-8 and TERC for grades 1-5). This is an opinion based on reviewing reviews of these other programs by people who have similar concerns with CPMP as do I. Other parents in Bellevue will need to make their opinions of TERC and CMP known. As for me, I plan to supplement my children’s 6-8 mathematics year-by-year until either they finish school or BPS faces the rude awakening that over $1 million was wasted on snap judgments to buy texts that almost no one in town wants. Perhaps the curriculum developers at BPS are correct that this is a good course for Bellevue students. If that were true, why were parents excluded completely from the discussion? (I volunteered to participate March 2, 1999 after the offer was extended by the school board. I was ignored completely.) From a political perspective, the BPS staff are employees and the school board are (voluntary) elected officials. If parents really do disagree with choices recommended by the staff and approved by the board, then parents will need to organize to protest. If you have found this, write-up, then you are already looking in one positive direction.
My review went more slowly than I had hoped. One reason for this was that I was asked to review the program as a whole and not to pick at one part or another because, say, I disagreed with the timing or placement of some topic. (An example: does it matter if the quadratic formula is done in the first year or not?) As much as I enjoy sarcastic asides, I have honestly tried to see how the program fits together and revise my comments as the picture filled in.
CPMP was developed with NSF (National Science Foundation) support and the cover letter includes all of the buzzwords currently popular with the NCTM (National Council of Teachers of Mathematics). In fact, the NCTM plays a major role in the funding and review process for CPMP and other NCTM initiatives such as CMP and TERC.
There are two reasons to read this. First, you hope to find something new about CPMP. Second, you are interested in honors mathematics education. As I worked through the four years of CPMP, I made notes of what I do and do not like. These notes are in separate documents (CPMP-1.doc through CPMP-4.doc for the four years of CPMP). This paper (honors math 2.doc) touches on CPMP only because CPMP is the example of the state of today’s mathematics educational materials which will be used in our community in the very near future. The real goal of this discussion is to place honors mathematics education in a context appropriate to students in the gifted programs and other honors students in Bellevue.
Here are my overall conclusions:
a) Both CPMP and CMP are wrong for honors students.
b) CMP seems to be wrong for everyone as a pre-algebra/algebra sequence.
c) CPMP looks like a decent enough non-honors or remedial course given that BPS wants a consistent program across the district that is currently in print.
We can gauge a given math sequence for quantity and quality.
a) Quantity: A list of algebra and geometry topics needed for competence.
i) Appropriate standards by grade level are needed. In Washington, all we have are the EALRs, a vague collection of goals set at grades 4, 7 and 10, and some frameworks which have an unclear position in the state.
ii) Standards for algebra, geometry, trigonometry and calculus have a purpose beyond getting a 500 or better on the Math SAT, passing the WASL or getting a 3 or better on the Calculus AB Advanced Placement exam. The correct purpose is to prepare a student to take calculus (or better) on entering college. (The structure imposed in Washington appears to be to assure that all students meet the minimum K-12 objectives required by law – and no more.)
iii) For honors students, the standards should be recognized as good by major universities with highly rated mathematics departments. At present, the 1997 California mathematics standards and the Japanese standards appear to meet this criterion while the standards (EALRs) of Washington do not.
b) Quality: There are several categories. These are somewhat more subjective.
i) The student materials should be self-contained (i.e., suitable for self-study).
ii) Definitions should be clearly stated and displayed.
iii) Key facts (theorems) should have elegant presentations (and proofs).
iv) Exercises and Problems
(1) Quantity of drill problems
(2) Quality of challenging problems – challenge problems should encourage a student to find new applications and new truths not discussed in the text.
(3) Major techniques used in proofs and derivations are reinforced.
(4) Answers to half the problems are in the back of the book.
v) Standard mathematical notations and terms should be used.
vi) The index should be comprehensive.
vii) Significant developments in the history of mathematics and significant mathematicians should be mentioned where pertinent.
c) Effectiveness: Do students thrive after completing these mathematics sequences?
i) Do student scores on standardized tests improve with the new materials?
(1) Scores from field testing are the usual basis for evaluation. Field testing, like pharmaceutical testing, has special concerns. There is a “placebo” effect in education just as in drug testing. In education, many factors influence class effectiveness besides switching the course texts. This includes special attention for the teachers with the new materials. There are special seminars and the teachers know that their comments and reactions will be used to improve the materials. Thus there is a pride of ownership that comes with field testing that is not present in standard implementations. Students usually perform better with a more excited teacher.
ii) Do students do as well in college math after the new high school program?
(1) How many take math in college?
(2) How many must take remedial math in college?
Parent involvement in textbook selection tends to be after-the-fact. Our first chance to look at a new book tends to be when a kid brings it home sometime in September. Usually, this is all right with most parents. After having CMP pushed at us in spite of objections and seeing an inadequate implementation, we need to pay close attention to the textbooks chosen. Between CMP for three years, TERC for five and CPMP for four with about 1,000 kids per grade and guessing about $70 per textbook, BPS has spent over a half million dollars just on math books within three years. I would like to believe that the money is being spent on textbooks worth having but the fact is that they are not.
The sections of this paper address a wide variety of topics. Here is the list:
A Bit on CMP (it is still less than we can chew and not worth swallowing)
What about CPMP? A general review with some specifics.
Why am I writing about all this stuff?
What about the History of Mathematics Education?
Observations on the last 5 years of math at Odle
How did math texts get to be so awful?
Overview of US math education, roughly 1875 to 1975
Overview of US math education, roughly 1975 to 1999
Child development and mathematics concepts – accelerate with caution
What is mathematics?
What is Algebra?
Higher level thinking in math
What should be in an honors course?
What can we do to fix this mess?
There is an important question that I keep avoiding. After thinking about CMP and CPMP, I need to state it as clearly as I can:
On what basis should you respect anyone’s opinion on how to teach your child mathematics?
We want to trust our schools to do right by our kids. I believe that most of the teachers, staff and administrators honestly believe that they are trying to do right by the kids. However, if the people making analyses and recommending decisions insist on asking wrong questions, then they will generally get wrong answers. The evidence shows that most educators are asking the wrong questions, at least for honors mathematics. Methods which work well for 70% to 90% of the population cannot be assumed to work when applied to 1% to 2% of the population, especially if that 2% is materially different from the mean of the 90%. Similarly, methods that may be appropriate for the bottom 15% of a population may give very wrong answers for the top 75%. This, too, is statistics.
One form of arrogance is for a non-teacher such as myself to presume to tell thousands of math teachers across the nation that they are wrong. Another form of arrogance is for math teachers to band together to tell us, in essence, that the needs of our children deserve to be ignored because they do not happen to fit the current educational paradigm. You will have to decide for yourselves whether to acquiesce to being cast adrift or to take arms against this sea of babble.
No one will agree with me completely. Most will disagree with me strongly on at least one issue. Here is the deal: convince me of your position and I will revise mine. Actually, convince me that your position has merit and I will try to incorporate it. If you convince me that it is important enough, I will distribute an updated discussion. Sometimes an open mind is full of holes but I still try to keep an open mind.
A Bit on CMP (it is still less than we can chew and not worth swallowing)
There must be something good to be said for CMP. Perhaps someone can enlighten me. I have yet to see anything worth preserving of CMP after reviewing over half of the units.
CMP may be a suitable vehicle for non-honors courses, although the numbers shown to the school board in May suggest that there is as yet no basis for such a conclusion. CMP appears to get a C+ for honors students and an F for non-honors students, based on its scores on its own process of evaluation. Since a minimum grade for successful completion of an honors math course is a B, CMP appears to be failing the honors students as well. There may be factors and interactions that may make the results of the early years of a program weak, as well as a limited sample of 50% of Bellevue students. Realistically, a stratified sample of 50% of a population (which seemed present) should provide very solid statistics. Unless there is a strong bias, these numbers should not be affected dramatically by more data.
Now, CMP was presented to us as having a very strong implementation program. This claim means that the teachers are not supposed to be the weak link. Blaming the students for not learning the material seems rather thin as an excuse if not simply inappropriate. That leaves the CMP material as the most likely source of weakness.
CMP for pre-algebra seems to be a series of disconnected pamphlets with lots of long, gory tasks which serve no identifiable purpose. CMP “algebra” avoids proofs, clear definitions, stated assumptions, rules of any sort and even clear results or theorems. This may be appropriate for non-honors mathematics (although I can find no rational justification). In an honors program, every effort should go to developing an appreciation of a field as understood by its practitioners and major consumers. After all, the honors student is likely to become a practitioner or consumer. By contrast, creative writing is one facet of an English course, yet it is not taught (I sincerely hope) in the complete absence of grammar, vocabulary and punctuation. Experiment and some understanding of scientific method is required in science courses. Yet in math, we see a program so stripped of mathematical content as to be intellectually crippling. Very bright students who have completed the CMP algebra sequence appear unable to retain any derivation more involved than transforming the equation for a straight line from one form to another. This is tragic.
The real tragedy lies in using CMP for honors students. The claims made for CMP included developing critical thinking and higher level thinking. CMP may actually succeed in this area, I cannot tell. The problem is that most students ready to take an honors program already have these capabilities developed further than CMP can take them. This is like using techniques to germinate seeds when the plant is already growing. For any of you who have done this, you know that the typical result for a plant is that the roots rot and the plant dies. Replace “plant” by “my kid” and I think you see my concern.
To put it mildly, the results shared with the School Board in May were “underwhelming.” Perhaps the year-end results were better but I am not holding my breath.
CPMP looks like a pretty good non-technical general math curriculum but still contains many disappointments. The overall goal of CPMP seems to be to expose students to as many applications of mathematics as possible and to many of the non-mathematicians who developed the applications. I do like this aspect of CPMP but there are problems with how they do it. CPMP describes itself as “Mathematics for Everyone.” That is, “one size fits all” is a design intent. CPMP was funded by the NSF (National Science Foundation) with guidance and review by the NCTM (National Council of Teachers of Mathematics). So now we can see our tax dollars at work in high school mathematics. (After all, it worked so well when several states legislated π= 3. One state actually set π=4.)
The fact is that CPMP is a tool and it may well be a good tool for the proper task. Using CPMP outside of its proper task can make a mess. (Have you ever thought about opening a can of soup with a ball-peen hammer?) CPMP has strong support for weaker teachers and is largely aimed at poorer urban centers with notoriously poor math. This is a wonderful target for improvement, a valid use of tax dollars and a good social goal. Now, is it Bellevue?
Students gain a superficial working knowledge of many topics but a good foundation in none. The variety of topics is really quite intriguing. There are topics from combinatorics, geometry, topology and even operations research (OR) that can really add to a student’s appreciation of the variety in applied mathematics. Unfortunately, each time I find a good or even elegant topic visited (briefly), there is another closely related but fundamentally less interesting topic which gets a great deal more attention and has the students do long and pointless calculations. I thought that this was stupid in 1967 (see the pre-calculus texts from the SMSG for really offensive examples) and I do not like it any better today.
CPMP finally states the notions I consider basic algebra – number field properties such as associativity, commutativity and distribution, among others –partway through the third year. A few properties are tossed in from time to time in earlier years (not many). Proof by induction shows up in the fourth year. The good news is that the fourth year (pre-calculus) has a lot of good content. The sad news is that the kids have to wait three years to get it. Something that really bothers me is that I learned over 90% of the material in the CPMP-4 pre-calculus course (both breadth and depth) as enrichment topics in my first year algebra course back in 1966-7.
As much as I like the variety of applied math topics in CPMP, the approach taken of doing lots of hard calculations is confusing. Worse, CPMP botches the abstract mathematics that it occasionally attempts. Most significant or fundamental formulas are hinted at in the student text but only printed in the Teacher’s Guide. CPMP is opposed to using clear definitions; assumptions are handled erratically (sometimes omitted and sometimes extra assumptions are thrown in that could easily be proved in exercises); and the few proofs that do show up in the third year are generally poor examples.
A proof is a poor example when the same fact or theorem can be shown a lot more directly and simply using the tools already at hand. This a huge problem in teaching mathematics: almost every theorem has a simple and elegant presentation – if only we are clever enough to spot it (or have the sense to remember it!). Most of the time people insist on doing things “the hard way” in elementary classes (i.e., calculus and below). In my experience, better mathematicians present things so clearly and simply that the whole development is a joy to behold (and a lot of it sticks). By contrast, poorer or less experienced mathematicians not only do long, horrible calculations they insist that these are necessary in presentations (and no one remembers much of anything). Long calculations are frequently needed in mathematics (abstract or applied), but better mathematicians know that this activity is not a spectator sport.
Most of CPMP’s long-winded calculations are related to topics that no one would do by hand any more. Actually, the silliest I have found had to do with finding the critical path in a project schedule. In the MORE problems after CPMP-2, Unit 1, Lesson 2, Investigation 4, the student is required to raise a 10x10 matrix to the 7th power in order to see that 7+8+8 is 23 (a fact obvious by inspection within seconds). While finding critical paths using matrices is a nice application of OR, it is a pointless calculation that no one should do by hand – especially when the question, “why does this work?” is ignored. This investigation mentions a connection with CPMP-1(where inspection was used to find the critical path). Contrasting the method in CPMP-1 and CPMP-2 leads to the inescapable implication that mathematics is developed to make the obvious harder to see. While that undoubtedly happens in some cases, the point to mathematics should be to solve problems in a way that makes sense and improves the quality of life. Done well, math can be elegant and insightful. Martin Gardner (columnist for “The Scientific American”) based his career on this. Help your kids read any of his books.
To avoid turning the mathematics into a “quagmire of formal definitions and theorems,” the CPMP designers just leave out definitions. This is explained with great force for vertex-edge graphs (teacher’s guide, CPMP-1, Unit-4, Lesson 1, Investigation 2) because the meaning of many mathematical terms is intuitively obvious to students.
The trigonometry in CPMP-3, Unit 1 really shows the problems that were built into CPMP by design and by intent. The CPMP-3 student edition describes some properties of right triangles on page 27 and on page 28 uses sin 29 and sin 53 without defining the sine of an angle. A few pages later, the text motivates the Law of Sines but states the Law of Cosines without proof. Over 200 pages later, CPMP-3 shows the Law of Cosines (a generalization of the Pythagorean theorem) using a geometric argument that is about nine times as complex as the first proof I thought of as an alternative. (See CPMP-3.doc.) The notion of “similar” figures (the geometric basis for trig functions) is introduced on page 298, or about three months later. Oddly enough, Hero’s formula for the area of a triangle takes about as much effort as CPMP spends on the Law of Cosines and shows the utility of algebra. Still, the student text for CPMP, Course 3 NEVER defines sine, cosine or tangent.
The math fundamentals in CPMP-3 really deserve our attention. At class session 65 (of about 170) of the math year, CPMP-3 begins to state the number properties in Unit 3, Lesson 2, Investigation 2. The lesson, titled “Algebraic Operations: Part 1” begins, “Music is a major form of entertainment in our society.” Seven pages later, the text states a few number system properties after this introduction:
“Mathematicians have spent a great deal of time studying arithmetic and algebraic operations to find basic principles that can guide work with symbolic expressions. They have generally agreed on the following basic properties of numbers and operations.”
The properties listed include the commutative and associative properties of addition and multiplication for (real) numbers and the distributive rule (stated separately for addition and subtraction).
The preamble all seems rather silly and misleading. The properties (with some duplicates!) form the definition a “number field” in abstract algebra. This is a convention – it is what the words mean. And, yes, mathematicians generally agree that the usual operations of addition and multiplication with the usual number fields (rational, real and complex) have these properties. Other number systems and operations may not have these properties. Are these the basic properties of the usual number system? That depends on what we mean by basic. I would not describe these as the basic properties for even the usual number systems for a general context, but they are suitable “basic” properties for high school algebra. If a mathematician prefers to study a different kind of number system (and many do), this does not reflect a disagreement with basic properties but rather one of scope. Not even the teacher’s guide provides any historical insight into the recognition of these properties or the algebraic structures common in modern mathematics. This is somewhat sad because the concept of a group (defined in my 1966-67 algebra I class) is quite useful in many scientific and technical contexts. With these definitions, the usual number fields are abelian (commutative) groups under addition and the non-zero elements are abelian groups under multiplication. Toss in the distributive law to relate addition and multiplication and we have a hierarchy of concepts to build the notion of a field using the notion of a group. Or do we not care about higher level thinking?
CPMP tries to fill in a broader mathematical background through the Teacher’s Guide. Frequently, this includes a quick description of the current status or past development of a topic. I find the discussions often vague or misleading – although rarely truly wrong.
CPMP often appears patronizing or condescending towards mathematicians. This apparent attitude in CPMP made me decide to see just how many mathematicians CPMP-3 mentions in the math index. Remember that CPMP-3 is the year they finally make contact with mathematical proofs. I found 6 names. Euler appears twice and his life and work rate some discussion. Venn diagrams have 4 entries, all pointing to the same example on one page. (The example shows the second most trivial example of the use of Venn diagrams and does not connect them to set theory. There is no discussion of Venn as a human.) The Pythagorean Theorem appears (again, no mention of Pythagoras or the Pythagoreans). The other three entries each rate at least a paragraph, and at least one has a picture. Interestingly, one is an economist, another a military officer and the third is a philosopher. None of these three made a contribution to mathematics. But their contributions do share one thing: they each invented a new view of VOTING.
Later in this document, I discuss my belief that the NCTM is collectively more interested in politics than in mathematics. As you (especially you math teachers) examine my comments, remember this statistic from the last paragraph: 75% of the people mentioned in the CPMP-3 math index are mentioned ONLY for political reasons. (Also remember that BPS is spending a lot of money on CPMP and CPMP is funded by the NSF, guided by the NCTM.)
In seeking to avoid a “quagmire” of mathematical formalism, CPMP has buried itself in rubble. There seems to be a Gresham’s Law at work here replacing the good coin of orderly sequence and logic with random leaps, unrelated and unsupported assertions and undefined terms. There is no progression of ideas or techniques. The only way a student can master the material is by rote memorization.
Possibly the worst feature of CPMP is the incompleteness of the student text. Students miss classes for many reasons. They take trips, get sick or even get tossed out of class. The textbook must state the critical information clearly and distinctly. I missed three full weeks of first year algebra due to illness. In courses with good textbooks, I returned to school caught up and in math I was about three days ahead. The CPMP (and the CMP) student texts are useless for self-study. This is a critical area where CPMP absolutely fails – by design and intent.
I can easily understand that you may find my observations, anecdotes and reminiscences on mathematics and its history to be deadly dull. Here I am trying to develop your insight into an esthetic of mathematics and I just pass you anesthetic. If it is any consolation, remember that I am the only person telling you about mathematics education and textbooks that is not either charging you for the comments, hoping to bill you $500,000 or building a career on the work. On the other hand, contributions are always welcome.
Perhaps the most important reason is that I take mathematics education very personally. One could say that first year algebra was the formative experience in my education. I tend to say that I woke up. I owe a great deal to that class and the teacher, Bill Leonard (at CSU Fullerton since 1974). This paper may be a payment on that debt. Or maybe I just like my own writing.
In the past 5 years, every time I have asked a math teacher, principal, curriculum specialist, assistant superintendent or superintendent in the Bellevue Schools why they allow or recommend texts with glaring deficiencies – such as failing to define the trig functions in the trigonometry chapter! – I have not gotten a straight answer. I am usually told that kids did not learn the material using the old methods or that I do not understand the full picture of the math sequence and things today are done in a different order and with different emphasis. 35 years ago, defining terms was important. I must be old-fashioned: I think that clear speaking and thinking begin with clear definitions and conventions. When someone refuses to define terms, it just makes me wonder whether they know what the words mean.
For years, I have argued with math teachers and administrators and even parents about the need for proofs in math. At the start, I was not all that concerned about proofs as such, I just wanted a “binary” issue (either you do or you don’t do it) unique to math. The more resistance I hit, the more I wondered about the issue of proofs. A course that does not prove any significant statements may be Engineering or even Art Appreciation but it is not Mathematics. As I saw the honors math course materials get weaker, I protested. This document began with those thoughts.
What makes a math teacher an expert on an honors math curriculum while my opinion is ignored? As near as I can tell, educators only listen to people (a) with degrees in education and (b) who are not parents in their district. The NCTM backs this attitude and the NSF has also adopted the view. Can all these people and organizations be wrong? Of course they can! Few of these people seem competent to tell an honors student how to learn math.
The concept of “expert opinion” is slippery. What makes an educator, even with a doctorate in education, qualified to tell you how your child should be taught mathematics? Your child is in an honors program and may be an identified “gifted” child so 98% of their training and research is irrelevant. Your opinion on honors math is likely to be more accurate than is theirs. After all, your child is in honors math. There is a much better than even chance that you or your spouse took honors math. Did CPMP experts take honors math? Can CPMP experts prove that they understand what your child should and can do? Earning an education degree does not give a person mathematical talent nor does it confer understanding of how smarter kids learn math. It helps a person learn how most people (do not) learn math. They may be experts for regular math but are they honors math experts?
What can make a person an expert on honors education? One way is to demonstrate both the mathematical aptitudes of an honors student and training and experience as an honors math educator. Most of the experts attached to CPMP show expertise as educators and have degrees in education. Few have even a BA in mathematics. Do any teach high school honors classes?
Searching for an objective measure that might convince us that they have merit and that we have merit leads to the standards: the SAT and GRE. SAT (pre-1974) or GRE quantitatvive scores of 700/800 or advanced GRE Math of 850/990 would suggest that the person is at least mathematically competent and may be qualified to advise us regarding our children.
Determining acceptable qualifications for framing an opinion on honors education has been completely overlooked. What do you think?
While many books are available on the history of mathematics, the history math education is not nearly as well known outside of professional teachers and professors.
I got interested in the topic when I was 15 and found my grandfather’s Analytic Geometry text from 1880. Along with my general interest in mathematics, I have accidentally studied a number of math textbooks printed over the past 150 years and observed dozens of teachers of math over the last 35 years. As a result, I probably know more about the development of math education than a randomly selected parent and perhaps even teacher but I am no expert. That does not stop me from describing what I have seen.
Most people discussing math education seem to be talking about how things are done correctly “today” and were done incorrectly in the past. For the past 15 years or so, a major component of correct mathematics has been political correctness at the expense of mathematical content.
This document provides background that may help you understand why I hold strong views on mathematical content and presentation. I hope that this will help you decide what you do and do not believe. The opinions are mine and I have stopped being concerned with politically correct sensibilities. When PC people use their politics as an excuse to harm my children they break the social contract. (And should be Locke’d away.)
Anybody can come up with a course outline for an honors math sequence. Jumping to the course outline avoids many issues that we try to ignore most of the time. I have been thinking about some of these issues since I was 15 and in some cases have even reached tentative conclusions. I will try to avoid “edu-speak.” I have used the term “edu-speak” for so long that for all I know I may have coined it. (Clearly, the term derives from Orwell’s 1984.) “Edu-speak” is an intentionally offensive characterization of the technical jargon of professional educators. Common words used by professional educators when talking with parents have technical definitions radically different from those in the dictionary. The technical definitions are rarely shared. A favorite example is “articulate.” Prior to discussions with BPS teachers and staff, “articulate” meant either a thing with moving joints like a skeleton (or a segmented bus) or a clarity and precision of speech. Of course, in education “articulate” and “articulation” are quite specific and refer to smoothing the progression of course content by subject when moving from elementary school to middle school or from middle school to high school. This is actually one of the least offensive and confusing examples of edu-speak I know. Perhaps a suitable group task for the readers of this document is to share other examples of buzzwords, buzz-phrases and psychobabble from edu-speakers.
The criticism of technical obfuscation goes back to Lilly, before Shakespeare, and got a big boost from Lewis Carroll’s Humpty Dumpty, “A word means what I choose it to mean -- neither more nor less.” At least one of the CPMP authors has read this – the quote shows up in a problem halfway through CPMP-3 – but the author appears to miss Carroll’s point. Humpty Dumpty is an arrogant buffoon who believes that commonly agreed upon definitions (conventions) are irrelevant. CPMP-3 tells us that formulating descriptions and definitions precisely as they will be used in an argument -–proof – is important to mathematical research. This is correct but incomplete. If the research is successful (it solves a problem), then the new definitions get close attention. There may be variations and finally a convention develops where the new usage or new term has a single widely accepted definition, at least in context.
Precise definitions, descriptions and conventions are critical in all walks of life – just look at a legal contract – and the point should be made nearer page 1 of CPMP-1, where it belongs. Neither more nor less.
Observations on the last 5 years of math at Odle
Is everybody happy? Perhaps the better question is: Is anybody happy?
We all seem to share a goal of effective mathematics education. Parents of gifted students believe that the math taught to gifted students should be effective for gifted students. Surprisingly, this view was not held by those selecting, implementing or even teaching courses.
Very recently, pre-algebra in Bellevue varied from school to school and textbook selection was “site based.” This led to many inconsistencies and perhaps even inefficiencies in the district. Selecting new math texts for elementary grades (TERC), pre-algebra (CMP) and high school (CPMP) may resolve many inconsistencies.
Prior to getting an honors math teacher, geometry (and up) students at Odle had to go to Sammammish High School in the morning. This worked academically but created a security risk as students walked unescorted between the two schools. Algebra and pre-algebra were taught in several ways. Some years, accelerated students had math taught by a gifted program (non-math) teacher and some years gifted students were in mainstream math classes. There were problems with both methods. When a gifted program teacher taught math, the problems tended to be those of a beginning teacher teaching math without prior training in the subject. When the gifted students were in mainstream math classes, the problems typically were social. 6th grade gifted students in a predominantly 8th grade math class had occasional social conflicts with older students. Teacher intervention was usually swift and effective so this was minor.
When the gifted program was able to bring in a math teacher for accelerated math courses, most of us hoped things would improve. For students outside the formal gifted program, access to accelerated math courses became available for the first time and no one had to walk to the high school because the new teacher also handled second year algebra. Unfortunately, the accelerated students found themselves in a classroom environment that was very poorly organized and only the geometry text had any value of the 5 levels taught.
A motivated student with a GOOD text can do well with occasional mentoring. The text needs to be self-contained and designed to support both self-study and class use. CMP and CPMP are both designed to require significant teacher involvement and many key explanations appear only in the teacher editions. These texts have been designed to prevent self-study, an inherently self-defeating approach for honors students.
The pre-algebra and first year algebra texts used prior to CMP were perhaps a bit stodgy and dry. Perhaps the teaching style of the teachers was somewhat didactic. As bad as this sounds at first blush, our personal experience was that both pre-algebra and algebra went smoothly and were well understood and received by our child (and all others where we have compared experiences). By contrast, we know of no one in PRISM actually happy with CMP. Here again, I am open to other comments, but please share details.
I am less critical of the Serra Geometry text only because plane geometry has never been simple to learn and many people are unclear about the goals. Serra’s text lacks foundations but does provide excellent enrichment and a decent geometric database. Adding rigor would be helpful but this is the least awful book we have had.
The Algebra 3 / 4 book is subtitled “A Graphing Calculator Approach.” Proof is ignored, definitions are few. Problem sets sometimes seem designed to show the student that a calculator is better at arithmetic than the mind -- implying that solving problems with one’s mind is a waste of time. The new mathematical content of this book is the introduction of the sigma notation for summing about page 192. This is optional and vanishes without a trace. This book should only be used as a doorstop.
“Technology in the Classroom” has become a new grail. As everybody seems to know, only good things come from spending several hours of math class each week working with calculators or computer software. Everybody except me, and perhaps thee, that is. As for me, I use and write software much of the time. Somehow, I have no difficulty separating this work from doing mathematics. I find that mathematics frequently helps me with software, yet software rarely helps me with mathematics – except in testing an idea or conjecture.
Perhaps some technology can help to excite the students to learn more math. What we see today is a focus on technology to the exclusion of mathematics. This may be fun for the teacher but it is far from clear how it helps the students learn math.
Harking back to the sixties again, I learned typing in a typing class. My English and Social Studies teachers preferred typed papers but no time in English or Social Studies was spent learning typing or formatting. In contrast, today’s educators believe that a significant percentage of mathematics class should be spent working with software packages on computers or calculators. The knowledge needed to operate a software package is far less than that required to do the related mathematics. Confusing mathematics with technology is extremely harmful to our children. Mathematics is about thinking while technology is about doing. Of course, they are related. They are not the same.
Technology is an important cultural development and may show good applications of mathematics. Technology can provide students important lessons in how people think and work together. This is more properly an example of the interaction of science, math and society. The James Burke series on the history of technology, “The Day the Universe Changed” is a much better forum for this overview than the math class. Some sociologists believe that the technology of double-entry bookkeeping for tracking business activities led to such increased profitability in early civilizations that its introduction was soon followed by a scientific and cultural revolution. This may have even sparked the Renaissance. This is a fascinating theory and belongs squarely OUTSIDE the math class.
How did math texts get to be so awful?
Overview of US math education, roughly 1875 to 1975
Actually, math texts today are generally a lot better than they were a couple of hundred years ago. It may seem silly to take such a long view, but it is important to have a sense of the history of mathematics in order to appreciate what the people writing books today are trying to accomplish and avoid.
Scientific and mathematical knowledge is learned through the efforts of people. People have attempted to create order out of a chaos of related facts to make that knowledge easier for others (and themselves) to learn. This has been a very long and frequently painful process.
Over the past three centuries, (western) civilization has democratized mathematics to an unprecedented degree. In the United States, we try to teach more mathematics to more students every year. This social goal is the basis for much that is good and a great deal of what is objectionable in textbooks today.
There were many changes in math education during the nineteenth century. The university system as practiced today emerged from a more communal and aristocratic structure that was still somewhat medieval in genesis. The development in the US paralleled or anticipated changes in Europe. It was sparked by the big-name universities following the American Revolution and echoed and amplified in the growing number of private and public colleges and universities across the US. The concept of a high school was also gaining popularity along with publicly funded primary (and secondary) schooling.
Many mathematics professors saw that incoming students could be better prepared if texts for mathematics bridged the gap between the arithmetic primers and college level books. Many late nineteenth century high school mathematics texts state this goal.
David Hilbert (of continuing mathematical fame) wrote a very well received revision of plane geometry around 1899. Hilbert’s approach provided a template for high school math texts for generations. Instead of blindly following Euclid’s Elements, Hilbert added many very reasonable postulates and applied the current notions of logical inference and proof. The result was a significantly updated and more quickly learned geometry. As symbolic logic emerged and abstract algebra became more widely accepted, Hilbert’s method was pushed down to algebra and the more formal T-chart proof common in geometry by the 1960’s (showing statements and reasons) emerged to dominate geometry.
For many years, that painful T-chart method of proof from geometry has confused me. Not because I found it difficult (generally, it seemed pretty simple) but because it seemed so cumbersome and detailed. Over the past 10 years, several experienced math teachers have told me that these proofs were dropped from geometry because students did not understand the proof method in geometry and were doing better with proofs in algebra. I accepted this until I saw that there were no proofs in algebra either.
After much reflection, it seemed to me that Hilbert was a better authority on mathematics than just about anybody. So WHY did Hilbert want us to do this stuff in geometry? Regardless of what may be written, it is possible that geometry may really be the secondary goal. The primary goal was to teach students how to reason carefully and correctly following simple rules of inference in a well understood environment. Nothing much pathological happens in the plane. A lot of bizarre mathematical facts show up in three and four dimensions as Hilbert knew (and helped discover), but the plane (especially with complex numbers) is well behaved. (“The reign insane falls mainly off the plane,” one might say. Or perhaps not.) Practical techniques and insights one has from carpentry or sewing generally turn out to be true. The game is one of proving these and finding new insights.
This change took awhile to catch on. My grandfather’s Analytic Geometry text (written in the 1880’s) seemed pretty advanced for high school back then and shows a love of “doing it the hard way” for no apparent reason. By the end of the 1950’s, the difficulty of reworking the elementary school texts to prepare kids for the high school college-prep math courses had been (ahem) resolved and “The New Math” emerged. Math teachers today still cringe when I ask about “New Math.” Tom Lehrer, a math professor and humorous songwriter commented on the New Math: “It’s so simple, so very simple that only a child can do it.” Among the problems with New Math were the inability of parents to help their children, too much math breadth for non-technical kids and poor to non-existent teacher training (that, of course, would never happen today – hah!). Teachers today refuse to discuss The New Math in any detail. However, many critics of the current NCTM offerings remember the New Math era and provide commentary and references.
How did math texts get to be so awful?
Overview of US math education, roughly 1975 to 1999
From “New Math” to “No Math”
By the time the New Math was seen as a complete fiasco, several changes had taken place. High school math teachers were organized and trying to improve their performance. This may well have been a response to bickering from college and university professors critical of incoming math students.
The NCTM probably formed in self-defense as many professional bodies do. The stated goals tend to address professionalism but there are often underlying reasons amounting to taking control of some power structure. The NCTM problem is a classic case of being stuck on the horns of a dilemma. Math teachers must prepare kids for college. Also, teachers must prepare those not college bound for life after high school.
For at least the past 25 years, organizations of teachers have taken on the task of defining public school math curricula. The NCTM has been dominant. This has translated to defining the math for private schools as well because of the economics of textbook publishing. In reality, it seems that Texas and California determine what gets published. Recently, the NSF began funding textbook development. This places the NCTM in the enviable position of recommending what will be learned in math and controlling, as an advisory panel, the purse-strings of development. The NCTM has become, in effect, the judge, the jury and the executioner.
Following the Civil Rights Movement, the tracking systems that evolved by the 1960’s were perceived as promoting social ills similar to discrimination on the basis of race. Tracking vanished along with many teaching methods that separated capable students from less capable students. Many teachers and principals today still believe that separating highly capable students from other students is a social evil. This mainstreaming trend also set the stage for the new textbooks. The re-emergence of honors courses in mathematics has created a contradiction for the textbook writers that ironically they are resolving with separate but essentially equal courses for honors and non-honors students.
The NCTM and other teacher groups are sincere in their belief that they are working to improve the delivery and content of mathematics courses. Interestingly enough, they believe that their statistics show this to be happening. The real problem is not with the quality of the teachers' efforts. The problem is with social goals that conflict with the needs of honors students. The good news is that teachers tend to be committed to working for the social good. The bad news is that their notions of social good tend to be cast in concrete.
The social goal in math has become the most math for the most students. This is a balance, of course. Most people will never learn algebraic topology or differential geometry. In fact, these topics are unlikely to be seen in any high schools. More down to earth, many first year algebra students have trouble with the quadratic formula so this has drifted to second year algebra. Proofs have drifted to the end of second year algebra (the third year in an "integrated" course) and pre-calculus. This drift has happened because the statistics show the teachers that many kids were not getting these abstract ideas and methods. The "social good" view of the math teachers seems to be implemented using this rule: NONE of the kids should be shown an idea unless a large percentage (which may be over 80%) of the students can handle it. This definitely led to a "dumbing down" in math. Even honors courses now delay important ideas until 2 years later than we saw them in the 1960’s. (Those of us that are that old, of course.) The dumbing down trend has accelerated in the past 10 years.
The Dolciani texts and other books like hers quickly replaced the New Math. Mary Dolciani was a teacher rather than just a professor and her series of mathematics texts became very popular. The MAA (Mathematical Association of America) has a prize in her honor. The Dolciani texts had a lot of discussion and examples, many exercises and a total disregard for professional mathematical terminology and notation. That last item is just an editorial complaint and I would probably swap CMP for the Dolciani counterpart in a heartbeat – given the choice.
Beginning in 1968, about every five years or so I have had a chance to look at a high school math curriculum. Whether I was just checking to see if I knew the material, trying to teach it (briefly) or just trying to save some friend's kid's bacon, I have looked at a full curriculum and usually found it adequate although rarely good. 35 years ago, I suppose that the new math permeated all pre-algebra and the math teachers of today are STILL screaming about it. After seeing the progression reach CMP, I believe that the teachers have now achieved "No Math."
There is a problem in distinguishing between problems of the book(s) and problems of classroom presentation. However, each unit of CMP that I have examined in any depth has shown similar problems. Important terms were used without definition. The glossary, if any, was inadequate and there is no index. Frequently new terms are use as if they are standard nomenclature when they are not standard terms. The students have only the pamphlet with the current unit so there is no way to refer to material covered earlier even though many pamphlets refer to previous units assuming that material is at hand. None of the exercises have answers in the back of the book.
In spite of my sarcasm, I support the apparent social goal of the NCTM of trying to deliver the greatest quantity of mathematics to the greatest possible quantity of students. The more I listen to how math teachers discuss the relative merits of math curricula, the more convinced I am that this view of “democratizing” mathematics is both devoutly held and worthy.
Unfortunately, as the social goal has been applied to students in honors courses, there is much to criticize. There is a basic assumption that one plan for a math curriculum fits all students and the only stretching needed is in the variety of homework problems (and perhaps some classroom exercises) and maybe an honors course can go a little faster. There is no difference in mathematical content, sequence, emphasis or real presentation.
Maybe the approach used in CPMP is appropriate for 75% of the population or more. There are some kids whose mathematical skills will never develop very far and perhaps no algebra sequence will be of use to these students. At the other end of the math spectrum, we have about 5% or more of the student population who are ready to understand and use mathematical reasoning if it can be presented coherently. The gifted population is entirely in this group, I think – even the kids who think that they are “bad at math.” The CMP and CPMP design and the design of most of the NCTM sponsored courses available today explicitly ignore the needs of the “right tail” of the student distribution but the designers and implementers refuse to acknowledge this limitation as a problem or even a topic of discussion. The result? The curriculum developers ignore us.
As math has been presented over the past few years to the 4th through 8th grade gifted students, the result is to make the kids do a prodigious amount of work designed to slowly develop higher level thinking skills that the kids already have. This is a waste of time for many and punishment for most. Besides, these kids need to have their thinking skills trained rather than merely developed. In repeated questioning over the past 5 years, I have found that training the thinking skills always happens “next year.” Remember Alice in Wonderland’s jam every other day? “Jam yesterday and jam tomorrow. But never jam today.”
Child development and mathematics concepts – accelerate with caution
Long ago, adults painted children as small adults and may have seen them as such. Today, this view seems ridiculous.
Piaget and other researchers in educational psychology tried to understand how an infant with no evident thought processes grows into a functioning adult (presumably with evidence of thought). Piaget focused on early childhood mathematical development. I suspect that this was because understanding concepts like numbers, addition and multiplication is much more “binary” than other subjects. (Either you know what “12” is or you do not. Either you recognize “2+3” as “5” or “3x4” as “12” or you do not, and so on.) Objectively observable concepts in child development are rare so Piaget became famous.
According to educators, further progress has been made since Piaget’s work but what I have seen just focuses on getting more detail about early childhood development.
There is an incredibly important change in a child’s thinking that allows algebra to work for that child. I remember this change happening for me perhaps because it was quite sudden as well as embarrassingly late. I do not mean to be mysterious. The mental change is just using a letter to represent a generalized number and beginning to manipulate letters as though they are numbers. This is the equivalence between arithmetic just with numbers and arithmetic using letters and then putting in numbers for a specific instance. Some people call it symbolic manipulation and substitution. …I call it algebra.
Once a kid gets the idea of using letters and numbers interchangeably, there is no real need to hold them back from algebra, unless the kid seems nervous. Before CMP, the accelerated 5th grade class started the year with a unit on the Pythagorean theorem. The unit gave the kids experience with selected triples like (3,4,5) and with A2 + B2 = C2. The essentially algebraic manipulations of solving for B given A and C were worked out in letters and in individual cases. This work was outside the regular math books so it may have been less scary to the kids. I saw several kids jump from arithmetic to algebraic understanding during this unit. As importantly, some kids saw the arithmetic but did not catch the algebra. This gave a wonderful check on proper math placement. Of course, there are kids in the accelerated programs who understand the idea of a generalized number before they get to 5th grade. The problem is one of recognizing this change and supporting the child appropriately.
The intellectual leap to a generalized number is the watershed for algebraic reasoning. I conjecture that there is a neurological component to making this leap as well as a learning element. A child should never be pushed into algebra (and perhaps not even pre-algebra) unless the wiring (neurological development) is ready for it. Historically, teachers have implicitly assumed that the pre-algebra sequence prepares the student for understanding generalized numbers in algebra. I believe that students’ brain development can vary as radically as any physical growth pattern (like teething or walking or potty training). Some kids may be physically ready for generalized numbers earlier or later than others yet nobody teaching them seems to ask this question. The use of the concepts following pre-algebra is more a coincidence of typical growth and maturity patterns and may not be due to the material taught. There is a standard fallacy in economics (and other subjects) stated, “Post hoc, ergo propter hoc.” (“After this, therefore because of this.”) Historically, pre-algebra and algebra have been taught to given ages because, coincidentally, those are the ages when the brain finishes a lot of final growth. Would any of the medical specialists in the crowd care to comment?
What is mathematics?
Sometimes people try to claim that mathematics is what a mathematician does. I suspect that my thoughts on the matter will land me in nearly as much hot water as that old sophistry.
Increasingly, I see mathematics as a game governed by rules. In “abstract” mathematics, we have primitive objects (sets, elements and numbers – just exactly what a primitive object is varies among mathematicians and students of math) which are described largely by analogy. Other terms (or ideas) are defined explicitly using the primitives. Axioms (assumptions) are stated and a logic (procedures for reasoning) described. Logic can be a study in its own right. I use a fairly simple “predicate calculus” or symbolic logic that is tied to truth tables. (This is not the only logic possible and even I have shown consistent logics with more peculiarly defined possibilities...I just do not use them in doing math.) Then we begin proving facts (theorems). We find new ideas to define and new facts to prove using these ideas. In “applied” mathematics, the goal is to find a way of describing some real world activity, event or process using a mathematical model. Physics is based on this. The world really exists outside us and we try to understand it. Newton developed one of the best known models for kinetics (moving objects) and his model stood until about 100 years ago. Newtonian physics is still quite accurate unless objects are massive, very small or moving at high speed. Applied mathematical models can be somewhat vague because a really good model gets confused with the reality we are trying to describe. Applied math is neither more nor less “real” than abstract (or “pure”) math but applied math is trying to model something real while abstract math need not. Most folks still seem to think that mortgage calculations are abstract rather than applied math.
What is wonderfully strange is that we can use a mathematical model to predict outcomes from new circumstances just by knowing the starting circumstances and mathematics. (This seems less true with chaos these days, but that is a misunderstanding. The problem with chaotic systems is that we cannot measure the circumstances accurately enough to make an exact prediction.) Good predictions do not make the applied mathematical model “true.” However, this does allow us to test the accuracy of the model and if the model proves accurate in all the tests, we start believing that the model is a true picture of the underlying reality. There is a caveat: nature reserves the right to surprise us. (“Terms subject to change without notice.”)
If the neat old exhibits at the Griffiths Park Observatory above Hollywood are still there, you can see a perfect example of the anatomy of a scientific revolution caught in hand-made polished wood and brass laboratory instruments. I last saw the exhibits in 1972 and they may have been destroyed by now, but I will never forget the overall impression. Along about 1905 to 1915, the Observatory acquired a collection of exhibits of physics experiments showing how light moves through the ether. The physicists in the crowd have already caught the joke: the Michelson-Morley experiments which discredited the “ether” view were conducted over 20 years before these exhibits were constructed. I loved the juxtaposition of this old exhibit hall with the visiting exhibit on the first walk on the moon. I hope that they kept the old stuff.
The point is that applied math is a really process of model construction and destruction. The goal is to try to find characteristics of (objective) reality which can somehow be modeled mathematically and then to deduce from the model. The major differences seem to be that in applied math, we concoct assumptions that link mathematical words to observable events or objects. We then turn the crank on the model using (abstract) mathematical machinery. With luck, we reach mathematical conclusions which we can convert back to statements about reality. Then – assuming we did the abstract math correctly and reality matches our model (closely enough) – the prediction should be pretty accurate. If not, build a new model.
Mathematics appears differently to each person who practices or uses it. This makes discussion of mathematics education frustrating. After a number of conversations with parents over the past several years, even we do not agree very well about what should be in a course for our students. Apart from the obvious view that we each think, “I am right and the other person is pig-headed,” what can we agree on for an honors course in math?
John Ruskin wrote, “The work of science is to substitute facts for appearances and demonstrations for impressions” [Stones of Venice, Vol. III, “The Fall”]. While this has been adopted as the motto of the Society of Actuaries, Ruskin’s words should ring true to anyone in mathematics or science. The Connected Math Program (CMP) seeks to substitute impressions for demonstrations. I worry that CPMP continues this “No Math” approach.
Determining what the goals of a math curriculum may be or should be is hard. This gets harder when we try to contrast what the kids get today with what we got 30 or more years ago. And we have a continuing question of what they really need. This is no different from the question David Hilbert and others faced over a century ago.
When I was in graduate school, I had a number of detailed conversations with engineering graduate students on areas of control theory (partial differential systems and differential geometry) and geology (topological vector spaces). As this was about 20 years ago and the engineers were just moving into these mathematical areas, I expect that a deeper knowledge of such mathematics is required today. These discussions showed me that engineers, financial economists and physicists do not wish to develop mathematics from the basement. They want to know enough about the fundamentals to be able to twist major theorems to their own needs. They need to know how to do the manipulations and to validate (prove) the appropriateness of their transformations and their base assumptions linking the model to their real problem. They tend to believe that full mathematical rigor is unnecessary unless the results look strange. However, they believe that if rigor is required that either they or their mathematical acquaintances can provide it. Published papers often show excellent mathematical rigor.
What do you mean, ‘Prove It’?
The fundamental notions in mathematics that mathematicians of all branches of mathematics will agree on probably boil down to this: we have some basic notions, some definitions, some axioms and some rules of logic (or inference) and after assuming that these things exist or are true we show that other statements are true.
Mathematics is about applying formal reasoning. Logic is actually a separate study which some view as part of mathematics and others as part of philosophy. Logicians may feel otherwise. Most mathematicians seem to use a fairly simple predicate logic about the same as what I learned in first year algebra. The logic I trust is simple and seems to be self-consistent and has so far worked (pragmatically) in all the mathematics I have had occasion to study. This does not make it “True.” About the best we can really claim is “not known to be false.”
In contrast, mathematical insight and intuition are not entirely logical. However, insight and intuition are trained using logical processes. The human mind is capable of wonderful leaps and connections in mathematical ideas just as in poetry or literature or other art forms. The difference in mathematics as a game or art form is that once having made the connection – seen the castle in the clouds, following Thoreau -- the rigor of mathematics requires us to build a foundation to reach that castle (or it is just vapor).
Viewing mathematics as a game of ideas can make it more fun to play – just remember to show the kids enough of the rules of the game so that they can play, too.
Even if we can just agree that mathematics is about using general rules for calculations, then it is a small step to accepting that mathematics is really about knowing how those rules are true. For something to be true, we must either assume it or prove it. The game of mathematics has been one of determining what we can (and cannot) prove based on specific assumptions.
There are numerous styles of presenting mathematics. For decades, US, German and French mathematics relied on an expository style that seemed low on motivation but high on content. Often a short discussion precedes axioms and definitions which are followed by theorems and proofs. British mathematics has often attempted to motivate theorems by placing a discussion including the proof before the statement of a theorem. Neither style succeeds in giving the student the sense of discovery that a mathematician feels when putting the ideas together for the first time. Sometimes the exercises really do help by getting the student to discover something on her or his own. Often they just help with memorizing the facts.
This “pennies from heaven” approach to mathematics exposition has been a problem for mathematicians since Euclid. In order to provide a coherent narrative with proper rigor, we usually lose sight of the process of discovery of a new idea. Frequently the problem is compounded by math “teachers” who know so little of the subject that they cannot defend logically any mathematical assertion the students may question. Such teachers fall back on their authority as teachers and rely on “proof by intimidation.” I tend to yell at people who use proof by intimidation. They have also violated the social contract.
Another approach of the incompetent is “arm waving.” With or without the gesture, this is evidenced by words like “obviously” or “clearly” although the speaker is unable to give any details when questioned. Something that looks true must be true. Similarly, something that looks false must be false. CMP appears to rely on arm waving almost exclusively.
Mathematics is full of surprisingly weird results which can really shake peoples’ sensibilities. One of my favorites is the Banach-Tarski theorem from the 1920’s. Given a three-dimensional solid, it is possible to cut it into finitely many pieces (very strange cuts!) and reassemble the pieces into a similar solid of different volume. Raphael Robinson at U.C. Berkeley showed that only 5 pieces (perhaps 4 – I cannot find a good reference) are needed to double the volume of a solid sphere. This is done with rigid motions (no stretching) and no holes or overlaps.
Educators today are fond of “ownership.” They seem to feel that a “portfolio” is evidence of ownership. In visual arts and writing, this may well be appropriate. However, to own a mathematical idea, a student must fully understand it. Ideally, the student should originate the idea by independent discovery and go on to prove it. Without a doubt, this is the best way to learn and remember mathematics. The problem is that a surprisingly large number of useful ideas have occurred to various people over the past 2500 years. It is difficult to re-discover all of them. Worse, few of us have 2500 years available. For purely pragmatic reasons, we cannot have each student reinvent all of mathematics, even though this might be ideal. Presented properly, math is a wonderful game but, again, we must remember to teach the kids the rules.
This leads to a different problem. Math is often studied as a just collection of rules -- rules useful for calculating answers to many quantitative questions but rules nonetheless. Mathematics is a very human invention and the student needs to know the game and not just the rules. Perhaps as important, the student needs to understand how people made up the rules and see why they can be fun -- and maybe how they can make up some rules of their own.
Many people ask "what is it good for?" about almost any statement from abstract mathematics. This is a great question for at least two reasons. First, any abstract statement in math may be nothing more than a mental exercise. This makes it sound rather useless. Second, many purely abstract developments in mathematics later become essential in physics and engineering -- although this may take decades or even centuries. So over time, these "useless" statements may become fundamental to our way of life. This makes mathematics critical to our survival even though any single result in math may well be completely useless at a given time.
What is Algebra?
Algebra began with solving linear equations in one unknown and has evolved a great deal since. Geometry has roots over 2500 years old dating back to classical Greece and the mid-East. By contrast, the concept of formally establishing a symbol for an as yet unknown number and deriving a value for the symbol from properties known did not emerge until around 900 AD.
The best definition of algebra I know is this: Algebra is the science of solving problems by using equations. (A science is an organized body of knowledge.) Math teachers today find this definition limiting because algebra “covers so much more now.” The “more” includes graphing, the data=function=graph paradigm, statistics, calculators, computers, software and other applications. Many of today’s math texts see statistics as the true beginning for algebra. Samuel L. Clemens placed statistics on a “truth scale” with this: “Lies, damn lies and statistics.”
The beginnings of algebra were not humble. The people developing the early techniques were often professionals working for the government of the richest (or soon to be richest) country around. This stuff – mathematics – was a key to expanding the kingdom by military conquest and in maintaining the power of a ruling aristocracy. Tartaglia is known in mathematics for developing the general solution to cubics (degree 3 equations in one variable) but Tartaglia’s day job amounted to developing ballistics for the artillery corps of his prince’s army. (The army with the better aimed cannon wins but CPMP does not mention this politically incorrect detail. During wartime, the man-in-the-street is singularly disinclined to stand still long enough for an explanation to a math problem. OR was developed during WW-II to hunt submarines.)
For the past two centuries, mathematicians have been increasingly concerned with proof and rigor. This got quite a bit of renewed attention about twenty years ago with the computer-based proofs of the Four Color Theorem and more recently with the difficulty of reviewing Wiles’ work on Fermat’s Last Theorem. Sometimes the push for rigor hid the beauty of the mathematics. In many cases, a proof is eventually found that is elegant and beautiful.
In a meeting to discuss mathematics books with the school superintendent, one of the questions I did not answer plainly was what book should be used for honors math. The best one I know for first year algebra has been out of print since about 1965. In fact, at the time of this meeting, I did not know anything more than the title and the publisher – and no one there remembers the book! (I also sidestepped whether I think I am a good mathematician. If I am, why did I become an actuary? Well, I could get a job. Oh, so I was not that good, right?)
In a related item, several parents asked for a more structured Math Club at Odle. This caused me to review many basics of high school math and think about what should be included.
At a May School Board meeting, I mentioned two items that I do not see in the CMP curriculum: unique factorization of integers and the quadratic formula. There are dozens more that I could have chosen. I felt that most adults with college education would recall these two easily. Most will recall learning the quadratic formula. Most will also recall learning about factoring, although they may not have seen a proof of unique factorization. A staff response (outside the boardroom) was that CMP-1 covers factoring and the quadratic formula does not need to be done in CMP-3 (or first year algebra). This response falls short. Factoring is introduced in CMP-1, but there is no proof or even a rough demonstration of Unique Factorization. There are several other facts which imply unique factorization. Yet kids finishing CMP do not know that if a prime number divides (evenly) the product of two numbers then it must divide at least one of the factors. Nor do they remember the Euclidean algorithm or how to find the greatest common divisor of two numbers. Deferring the quadratic formula to second year algebra has become a way to ignore it there as well. Can Algebra 3 / 4 graduates derive the quadratic formula? I hope parents of Algebra 3 / 4 students will quiz their kids and let me know.
Back in ancient times, when I took first year algebra, the quadratic formula was a big deal. I think that we spent several weeks on the topic. (I was out sick for three weeks during the development and that gives me some gauge on the time frame.) After working through expanding lots of squares like (3y-4)2 and (ax+b)2, we moved on to completing squares (e.g., if you add 9 to x2 + 6x you have a square). Finally, we used completing squares to derive the quadratic formula. The test on this unit was to do a very detailed, step-by-step (with reasons) derivation of the solution. Unique factorization was done much earlier in the year and in as good detail as in many college level abstract algebra texts. (“Do it right or skip it.”)
This is very different from an introduction to primes in which a few numbers get factored or a lecture which presents the quadratic formula for rote memorization. Please understand that I think students should learn about primes and factoring in pre-algebra (or earlier) by examining specific examples. This is background for appreciating a general fact. I also have no quarrel with a lecture in an algebra course on the quadratic formula. My dispute centers on the students not learning how to derive the quadratic formula and not understanding proofs of any general facts about integers or polynomials.
A surprising question
In a discussion, one BPS curriculum developer asked me:
“Why do first year algebra students need symbolic manipulation?”
I still feel this way. The whole point to algebra is symbolic manipulation. Everything else is applications. The graph=function=data paradigm is very useful, but to me it is still just a sort of application (although a good case can be made that it is a generalization).
I will try to state a view that I just plain do not believe has any truth to it. I must apologize in advance if I get it wrong. I will be happy to distribute a correction or clarification. The point seems to be that…
Math teachers have found that first year algebra students generally do not understand symbolic manipulation and that the brighter students can hide a lack of real understanding behind mere dexterity.
Analyzing this, I see two items. First, students who should be in a non-honors course or perhaps need to mature physically (that is, neurological development has not yet reached the critical point for algebra) do have trouble with symbolic manipulations. Second, it seems that math teachers have trouble figuring out which things the smarter kids do not understand.
I do not see why either item is a concern in an honors course. The first is a problem of selection or timing. In a class of uniformly brighter kids, it should be easier to spot the problems that some of them have. Anyone aware enough of the mental processes in math should be able to gauge the readiness of a student for symbolic manipulations. I think that most parents can probably assess their kids as long as they are honest with themselves.
Why learn symbolic manipulations?
I was then asked for further information on why students, specifically honors students, really need to be able to do formal manipulations and proofs. That convinced me that we have a very serious difference in opinion as to the purpose of an algebra course. To me, this is the same as asking why students taking French should learn French vocabulary and grammar. And the answer is: so that they can solve problems appropriate to their course level and, in the long run, do well in math. Alternatively, so that they can compete effectively against other math students who are not being fed watered-down crud. My opinion is viewed as irrelevant.
I do have a number of specific tools and techniques that I have used ever since I learned them in the early days of symbolic manipulations. These two examples use methods that I learned along about October of first year algebra. I think a first year honors algebra graduate really ought to be able to do the first problem and understand the second well enough to complete the solution sketched here. I use algebra to simplify arithmetic. It seemed the more I learned to apply algebraic reasoning to arithmetic, the faster and more accurate my arithmetic became.
1) Consider multiplying 9,875 times 10,125.
Most folks probably head for a calculator and punch in the numbers. But this is a mental math problem. First it is a difference of squares:
9,875 * 10,125 = (10,000 – 125)*(10,000 + 125) = 10,0002 – 1252.
To square 125, first multiply 12*13 = 144+12 = 156 and tack on 25: 1252 = 15,625. Then subtract this from 108 to get 99,984,375. The trick is to get the right number of leading 9’s.
The difference of squares: x2 – y2 = (x+y)(x-y) solves more problems in mathematical competitions and timed tests like the SAT and GRE than any other single formula. This is why I focus on the difference of squares so strongly in Math Club. It is also great for multiplying.
The other formula is for squaring numbers ending in 5: (10x+5)2 = 100x(x+1)+25.
A human mind is better than any calculator but the mind needs to be trained. How real or valuable is this sort of skill? I suppose that depends on whether you want to see your kids get into the colleges they want.
2) This problem is from the 1999 Invitational (the qualifying level for the USA Math Olympiad). What is the sum of all integers x such that x2 – 19x + 99 is the square of an integer?
This solution is more complicated and I am just going to sketch in the main ideas. After a few false leads, I tried to solve a (Diophantine) equation for x and y integers: y2 = x2 – 19x + 99. After completing the square on the right, multiplying by 4 (to clear fractions) and factoring the difference of squares, we have
(2x-19+2y)(2x-19-2y) = -35.
This writes –35 as the product of two integers (because x and y are integers), so we can separate the quadratic problem into 4 cases of two linear equations. The 4 cases come from the distinct ways of writing –35 as the product of two integers (y non-negative): 35x(-1), 7x(-5), 5x(-7) and 1x(-35). Since y>0, 2x-19+2y > 2x-19-2y. To find one value of x, we set
2x-19+2y=35 and 2x-19-2y=-1. Add these to get 4x-38=34 and solve for x = (34+38)/4=18.
Finding the four values of x which work (1, 9, 10 and 18) is not terribly difficult by trial and error. But how do you know you have found ALL the values that work? CMP will not help you, nor will your calculator. CPMP students will not get this before pre-calculus, if at all. You need a proof and you need algebra courses that focus on mathematics fundamentals rather than group task skills and memorized applications.
Higher level thinking in math
“Higher Level Thinking” and “Critical Thinking” are two of today’s great edu-speak phrases.
The notion of higher level thinking seems fairly reasonable. We represent a collection of ideas and facts as a single object. Then we use that object as a building block for a new mental construct. For example, “democracy” is a high level concept which includes notions of government, a collection of documents, a history spanning several civilizations and so on. Each person’s notion of democracy is different but we share many common components.
Mathematics is based on forming higher level concepts. Each definition and theorem adds to an idea. For example, a “group” is a set (of things – numbers, functions, actions) together with an operation (a rule which takes any two elements of the set and gives us a unique result in the set) satisfying a few properties. There is an identity element; for each element in the group there is an inverse; and the operation is associative. The integers with addition are a group: 0 is the identity and the inverse of a number is its opposite [i.e., a + (-a) = 0]. Addition in integers is associative. With just these properties, we can prove a large number of facts about groups. Why care? Groups show up in many settings in physics, chemistry and even some models of biological systems as well as in abstract mathematics. You just learned a high level concept.
Critical thinking seems less well specified. Apparently, this means that a person can form a consistent chain of inferences leading from things that are known to a statement that one would like to show is true. If this is even nearly correct, how does this differ from proof? Words are a lot more slippery than the logic used in algebra. Most people cannot reason consistently and correctly using words for very long. A standard college game is to try to talk people into contradicting themselves on matters that they believe to be true. Most people can be talked into a contradiction within 15 minutes. The logic used in algebraic and geometric proofs is simple and easily believed. This logic is a model of the logic used in non-mathematical situations. In real life, we rarely know all the factors and we are often unaware that we have hidden assumptions. As a result, most people do not reason as much as jump from one statement to another based on emotional conviction. The fact that the conviction is emotional makes the jump immune to rational analysis. (This is an observation not a criticism. Much important work must be done with incomplete information and intuitive leaps. The problem is confusing an intuitive leap with logic.) Skill in critical thinking should allow a person to spot when a jump is rational or intuitive and to recognize hidden assumptions.
People differ in thinking ability. This statement is probably not in current political favor but it would get pretty dull if we did not differ. Denying differences in thinking is as absurd as claiming that there is no general difference in learning styles between males and females. Different does not mean better or worse. Fairness is less likely to follow from denying differences than from recognizing and understanding different needs and abilities. In mathematics, topologists, analysts and algebraists all think quite differently. Many of the more significant breakthroughs of the twentieth century have come from discovering that their views are complementary. This is synergy at its best.
Honors students typically have higher IQs, the selection criterion favored by BPS and many school districts in its gifted programs. Such students are likely to develop higher level thinking on their own. They may not need – and in fact may suffer from – a program that attempts to build higher level thinking from the ground up. Instead of digging intellectual ditches with spoons – a translation of the approach used in CMP and much of CPMP – students should be learning to “stand on the shoulders of giants” (Isaac Newton). CMP and CPMP supposedly promote higher level thinking and critical thinking but the building blocks are absent. The claim is empty; the cupboard bare; and the emperor has no clothes. Good definitions and explicit logical connections are required to make the process work.
What should be in an honors course?
The goals of a math course need to be determined before a curriculum or a text is selected. In a high school honors course, material should be chosen and presented to prepare the student to use the material in college either as a major or a prerequisite to a major. While this might be a purpose of AP (Advance Placement) courses, AP courses are really the last stage of the process. Once we agree on content, the text should be sufficiently self-contained that a motivated student can learn on her or his own. The NCTM and I disagree completely on this.
Mathematics includes a language, a craft and an art form. Most people stumble over the language, get stuck in craft details and never see the real craft or the art form. CMP and CPMP both show this pattern although CPMP does try to show examples of applying math to other subjects. The math textbook is a student’s first – and often only – contact with mathematical methods. Making that contact good may require different books for different students.
We need to ask what future physical sciences, engineering or mathematics majors need from a high school curriculum and compare this with the needs of students planning to major in less technical fields. Any list tends to be fairly lengthy because people tend to focus on items in a curriculum very quickly (as in, “When should we do the quadratic formula?”).
To address the quantity required in a math sequence, “Little Read Book.doc” contains a list of topics that I would love to cover during Math Club next year. This list is a fairly detailed course outline for algebra (first and second years) and pre-calculus. The geometry content seems a little weak, perhaps because I tend to view geometry from an analytic perspective. I call this the “Little Read Book” partly to poke fun at how my advice on studying mathematics gets ignored but mostly because we all want our kids to know math but we avoid actually thinking about the process. The “Little Read Book” does not address quality issues except indirectly in the reading list. For now, some comments on quality are in the various CPMP reviews.
Honors math students need to know a great deal about symbolic manipulations. They should be able to transform verbal problems into equations and solve the equations accurately. They should be able to solve a wide variety of problems. In fact, once they have finished first year algebra, they should be able to do all of the problems on any AJHSME exam (but without a time limit). Once they finish second year algebra (or the third year of an integrated program), they should be able to solve any problem on the AHSME (again without time constraints) and do every problem on the SAT quantitative exam. The only difference between the math SAT score a student gets and 800 should be due to the student’s speed or psychological response to the testing environment. The student should not be ignorant or unskilled. As a reality check, MENSA required a 1300 total SAT score (pre- 9/74). This translates to roughly 700 or better on the SAT technically directed people. (650 on the old scale, plus the 50 point rescale.) For gifted kids, this level of math ability is a minimum threshold that all of them should attain unless actively prevented by their schooling. People are born to learn and gifted kids more so. We can interfere with that process and maybe even halt it. Or we can find ways to encourage it.
The topic list in the “Little Read Book” comes from three sources. Some topics are from special lectures or seminars, like NSF math enrichment summer courses for high school students (discontinued in the 1970’s, I believe) or Brother Alfred’s talk on Pick’s Theorem at a conference for high school students. [Brother Alfred was then chairman of the math department at St. Mary's College in Moraga. In about two hours, Brother Alfred developed and proved Pick’s formula using a high degree of audience participation. One 1997-98 Algebra 1/2 class spent most of December discussing but not proving Pick's Theorem.] Most of the topics are from the first year (honors) algebra course I took in high school. All in all, we could do a lot worse than to clone that mid-60s course. Despite good intentions, BSD following the NCTM has done worse for our kids so far, unfortunately.
What can we do to fix this mess?
Developing insight is important in math. Developing the craft of mathematics is just as important. The current honors syllabus attempts to motivate mathematical concepts and make the concepts interesting by relating the ideas to real world situations. As upbeat and promising as this seems, the depth is so shallow that only trivial intuition will develop. The student must memorize all formulas and develops no analytical or reasoning ability. Deeper insights in math require some contact with the language and craft of mathematics. The craft is embodied in the proof. Many ideas in mathematics come from humble beginnings such as drawing pictures or lengthy calculations. All are subject to the same requirement: no statement is “true” until proven and the proof reviewed by other mathematicians.
About the only options we parents seem to have at present are:
1. Supplement our kids’ math with something coherent and logical.
2. Build a consensus on the needed honors math content and recommend materials.
3. Find a new school district (i.e., move or go private).
4. Home study mathematics.
5. Give up.
6. Change the state level mathematics standards to include mathematics. (This will filter down to district and school level changes quickly.)
At this point, I am trying to supplement. At worst it cannot hurt. Unfortunately, the CMP and CPMP programs can cause a great deal of damage if not stopped. The homework assignments are very long and seem to serve little purpose. As a possibility for a change, we might just eliminate CMP and see how the kids do jumping from 5th grade math to CPMP-1. As experiments go, it might be worthwhile. (And it is unlikely to be worse than CMP.)
Finding better tools, at least for honors students, seems to be the right answer. However, unless we parents have a fairly clear picture of what we believe is needed, nothing will happen. The odds are that nothing will happen if we do come to a consensus but at least we will have a basis for negotiation. Especially if the early CPMP benchmarks are as dismal as those of CMP.
Currently, education specialists focus on the outcome of the learning experience: a student should be able to explain his or her solution to a problem to the “man in the street.” This is interesting but the implementation contains a flaw: are the specialists looking for the right outcome? An honors math student has an important audience that is completely overlooked. The honors math student must be able to explain the answer to her or his professor at a university or even to a practicing scientist or mathematician without appearing completely ignorant. As odd as it may seem, I have never found a “man in the street” who cared that the alternating group on 5 or more items was a simple non-abelian group. One of my professors was quite pleased when I displayed an original proof at the oral final – and waived the final!
So you see, we really do need to consider the appropriate presentation for the audience.
I look forward to your reactions.