QUESTIONS FOR THE NATIONAL MATHEMATICS ADVISORY PANEL

Location of this page: http://howothersthink.com/nmap.htm


2006.12.3 Sunday

Difficult questions may elicit deep answers, so the panel will get better at what it really wants. 

1. The Usefulness of Mathematics
2. The Crisis in Mathematics Education
3. Truth to be Told to a Benevolent Despot.
4. The Structure of Educational Governance
5. Treatment of the Gifted
6. The Panel as a Sham
7. Taboo Issues								

Appendix 1: Charter of the National Mathematics Advisory Panel
Appendix 2: I Samuel 17 

The Duchess's Epilogue


Disclosure: I was a math major as an undergraduate and have read on my own a good deal about logic, set theory, metamathematics, and foundations. I can hardly be accused of not liking the subject. 

QUESTION 1. THE USEFULNESS OF MATHEMATICS (three articles)

ARTICLE 1. WHETHER MATHEMATICS IS USEFUL?

It would seem that mathematics is widely used.

Objection 1: Mathematics is mostly useless, except to those very few who will become active scientists and engineers. Engineers use mostly algebra, a very few formulae in geometry, and rarely calculus. For the rest of us, not even algebra gets used. When I tried to put some simple equations into something to be read by political appointees, I was told to take it out, it would not be understood. It would have been very nice the time I represented the policy unit at some technical discussions about regulations if the lawyers knew basic set theory. I wanted to interrupt and get them to write out some simple set formulas rather than long-winded phrases.

Objection 2. Even in the sciences, thinking is rarely as exact as it is in mathematics, and engineers rest content with good rules of thumb. Going down the ladder, the reasoning of advertisers, politicians, preachers, and lawyers is horrendous. Deirdre McCloskey told me a few months ago that Donald's estimate that a quarter of GDP is devoted to persuasion should probably be increased to 30 percent. Out with Euclid's Elements, in with How to Lie with Statistics and The Art of Cross-Examination.

On the contrary, the Panel should ask businessmen to specify just what they want, both for lower math skills for the bulk of their employees and for those who will use math beyond junior high school.

Reply to Objection 1. Employers will know what skills they really want, though they need to articulate them far better.

Reply to Objection 2. It is important that students realize what exact reasoning is, the better to compare it with inexact reasoning and bogus reasoning. Learning  mathematics is essential to this goal.

ARTICLE 2. WHETHER THE NATURE OF MATHEMATICS THINKING IS UNDERSTOOD?

It would seem that we know generally enough about the general principles of proofs, formulae, sets, and so on to get on with the business of instilling the habits of exact reasoning that characterize mathematics.

Objection 1. It would seem that attempts of specify more exactly just what mathematical thinking consists of are failures. It is not enough to just teach the same old math over and over again, but to envision what basically is at foot. Such pronouncements, like the one below, are circular and not helpful.

http://www.qsa.qld.edu.au/yrs1to10/kla/mathematics/ppt/trw_mathematically.ppt
Here are the high points: 

What is thinking mathematically? 

* making meaningful connections with prior mathematical experiences and knowledge including strategies and procedures
* creating logical pathways to solutions
* identifying what mathematics needs to be known and what needs to be done to proceed with an investigation
* explaining mathematical ideas and workings.

What is reasoning mathematically? 

* deciding on the mathematical knowledge, procedures and strategies to use in a situation
* developing logical pathways to solutions
* reflecting on decisions and making appropriate changes to thinking
* making sense of the mathematics encountered
* engaging in mathematical conversations.

What is working mathematically? 

* sharing mathematical ideas
* challenging and defending mathematical thinking and reasoning
* solving problems
* using technologies appropriately to support mathematical working
* representing mathematical problems and solutions in different ways.

Objection 2. Furthermore, there is there is a small subfield in education called "transfer of learning," the idea is that learning one subject transfers to other subjects. Near transfer is algebra to geometry or algebra to physics. Far transfer is what my English teacher said when I asked him why we were reading fiction, that is, books about things that were not true. "To learn about life!" he said. I now agree that novels can get at human nature in a way that biological and social scientists cannot. Far transfer is about Latin or geometry or, well anything, that teaches one how to think.

In fact, little is known about the transfer of knowledge of mathematics, specifically, to other fields.

On the contrary, while not nearly enough is known about the nature of mathematical thinking and the transfer of that thinking to other fields, our ignorance is not total. Accordingly, the Panel should dwell upon this issue of transfer.

Reply to Objection 1. This will not do! There's an anthology collected by Robert J. Sternberg and Talia Ben-Zeev, edd., The Nature of Mathematical Thinking (Mawhaw, NJ: Lawrence Erlebaum, 1996). See the review by John Mason, 'Describing the Elephant: Seeking Structure in Mathematical Thinking,"  Journal for Research in Mathematics Education, 1977 May. The title gives the gist of the review, but the book's chapters should contain ideas for the members of the Panel. Furthermore, informal characterizations of how mathematicians think can also be illuminating.

Three men with degrees in mathematics, physics and biology are locked
up in dark rooms for research reasons.

A week later the researchers open the a door, the biologist steps out
and reports: 'Well, I sat around until I started to get bored, then
I searched the room and found a tin which I smashed on the floor.
There was food in it which I ate when I got hungry. That's it.'

Then they free the man with the degree in physics and he says:
'I walked along the walls to get an image of the room's geometry, then
I searched it. There was a metal cylinder at five feet into the room
and two feet left of the door. It felt like a tin and I threw it at
the left wall at the right angle and velocity for it to crack open.'

Finally, the researchers open the third door and hear a faint voice
out of the darkness: 'Let C be an open can.'

And this:

An engineer, physicist, and mathematician are all challenged with a
problem: to fry an egg when there is a fire in the house.  The
engineer just grabs a huge bucket of water, runs over to the fire, and
puts it out.  The physicist thinks for a long while, and then measures
a precise amount of water into a container.  He takes it over to the
fire, pours it on, and with the last drop the fire goes out. The
mathematician pores over pencil and paper.  After a few minutes hegoes "Aha!  A solution exists!" and goes back to frying the egg.

Sequel:  This time they are asked simply to fry an egg (no fire).  The
engineer just does it, kludging along; the physicist calculates
carefully and produces a carefully cooked egg; and the mathematician
lights a fire in the corner, and says "I have reduced it to the
previous problem."

These and many more from http://www.xs4all.nl/~jcdverha/scijokes/6.html. Go ahead and indulge yourself. Keep going with 6_1.html and 6_2.html.These jokes should inspire some thoughts, not that mathematicians come off best, but that you might wonder (the beginning of wisdom, recall) just what is to think like a mathematician.

Reply to Objection 2. Transfer of knowledge is certainly an important issue. Shifting students from useless to useful math courses will accomplish far more than all manner of improving teaching methods for useless courses. But our ignorance on this issue is not totally bleak. It's just that so little is known, esp. at the K-12 level, about transfer of knowledge, esp, from math to far fields.


ARTICLE 3. WHETHER GEOMETRY IS AT ALL USEFUL?

It would seem that learning the method of rigorous deduction is useful to all in evaluating arguments of all sorts.

Objection 1. Geometry does indeed teach the art of making rigorous deductions. (Forget that Euclid did not know that, if b is between a and c, the b is between c and a.) The fact is that deduction is not all that rigorous in physics. (What is the event space in which special relativity operates? It is not a metric space, for two distinct events, a photon leaving the sun eight minutes ago and its arrival on earth now has a zero Minkowski metric. I could not find an answer in the physics library when I was an undergraduate math major at U.Va. and had to await Mario Bunge's Foundations of Physics (1967), from which I have lifted the first sentence of the Duchess's Epilogue. Even so, most physicists pay little attention to lack of rigor.) 

Objection 2. Geometry is little used even by mathematicians. It is enough for scientists and engineers simply to know various formulae, like the Pythagorean theorem, which can be taught quickly using algebra, and not burden them with a year long course in geometry, which comes at the expense of studying probability and statistics. Knowing how to spot bogus statistical arguments is helpful to everyone, not just those few who will ever use the theorems of geometry.

On the contrary, teachers should continue to acquaint students with rigorous reasoning, though not necessarily through geometry. The Panel should ask how this acquaintance might be accomplished more effectively and efficiently. A balance should be struck between the conservative principle of retaining the wisdom of the past (which includes the teaching of geometry) as opposed to Mr. Jefferson's "dead hand of the past" and Mr. Mencken's definition of tradition as "the cumulation of centuries of imbecilities."
Reply to Objection 1. Deduction isn't always so rigorous in mathematics. Recall the ghosts of departed quantities, abolished by Bolzano and Cauchy (see ) in the nineteenth century and reinstated rigorously by Abraham Robinson in the 1960s.

Law is much, much worse. Get your students to read some Supreme Court opinions. IF you have gifted students and IF you are a highly gifted teacher, your students will discover that these opinions fall far short of the standards of rigor of geometry. How can such learned judges come to opposite conclusions or issue concurring opinions? We know what the Court actually decided (except of course that future courts will have to interpret the decision). It is useful to know that the law is much less rigorous than geometry (except that all those who have suffered both geometry and law courses don't seem to know it).

Reply to Objection 2. While perhaps an entire year of geometry now comes at too high an opportunity cost of teaching probability and statistics, experience with the "New Math," basically the use of the axiomatic method for algebra shows that geometry is a far better and more proven way to acquaint students with the method of rigorous deductive thinking. Trigonometry has largely been eliminated as being too costly, and so geometry might be scaled back also, but a working experience with the deductive method is too important to forego. (Admittedly, just what the transfer of knowledge to near and far areas consists of is understood much too poorly.)

This Panel won't recommend scrapping math beyond the eighth grade, but at least ask what would happen if students no longer had to suffer from high school math. (Why is school so boring? Solve this, and the education problem in the country is licked!)

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QUESTION 2. THE CRISIS IN MATHEMATICS EDUCATION (four articles)

ARTICLE 1. WHETHER THERE WAS EVER A GOLDEN AGE OF LEARNING?

Objection 1. It would seem that there never was a golden age of learning, and students are doing about as well as they ever did. The legend about an inordinately difficult eighth grade 1895 test in Salinas, Kansas, is either bogus, not what it is claimed to be, covers a select population, or misinterpreted. (Use Google on this. Quite illuminating.) I did not find any long-term studies for mathematics, but Sam Wineburg's delightful, "Crazy for History," The Journal of American History, 2004 March, argues this to be the case for American history, at least since 1917, and Dale Whittington, "What Have 17-Year-Olds Known in the Past?" American Educational Research Journal 28(4) (1991): 759-80, details specific tests. (I can supply the articles.)

On the contrary, whether there was ever a golden age of learning, today's economy demands better learning than existed in the past.

Reply to Objection 1. Still, there has been a decline in test scores starting in the 1960s, and this must be addressed.

ARTICLE 2. WHETHER ANYTHING NEEDS TO BE DONE?

Objection 1.  It would seem that the normal forces of supply and demand would ensure that the numbers of mathematicians and yoga instructors would be set by the market. If the demand for mathematicians should rise, the number of students majoring in the field would also rise. There are no laws limiting the number of courses in math one can take or the number of math majors at a college.

On the contrary, there are certainly many ways the education system does not work properly. It is not the failure of higher education that is at issue but insufficient numbers of those prepared to profit from studying mathematics after high school. The Panel needs to clarify just what failures need to be addressed and how.

Reply to Objection 1. There are at least three kinds of failure at work

A. Market failure. One reason there are public schools is that too many parents do not meet the economist's criteria for rationality and that the public wants to protect children from their irresponsibility. Furthermore, we all tend to have short time horizons, optimal perhaps for our hunting and gathering days, but suboptimal now. In implicit recognition of this, voters regularly elect politicians to cope with this suboptimality by mandating forced savings for adults and compulsory education for children.

B. Government failure. Teachers' unions make it mandatory that math teachers get paid no more than English teachers. There is a shortage of math teachers, since they command a larger salary in the market. This is failure at the State level, failure to reign in nation-wide rent-seeking by unions. The President introduced legislation to cap medical malpractice settlements. The Democrats, who get the lion's share of political contributions from the National Trial Lawyers Association, blocked the law by filibustering in the Senate. The No Child Left Behind Act, by contrast, went through, due to the extra monies promised to the schools, more than enough to make up for hypothetical withdrawal of Federal funds after 2013/14.

C. The abiding failure of human nature. The problem could be as old as when an animal could first explore and learn from its environment and so was no longer dependent on rigid genetic instructions. Perhaps in the Old Stone Age, when our basic thought patterns were set, children learned everything their parents wanted them to. Certainly by the Bronze Age, this was no longer the case, when the Lord Himself had to mandate instruction:

Deuteronomy 11:19. And ye shall teach them your children, speaking of them when thou sittest in thine house, and when thou walkest by the way, when thou liest down, and when thou risest up.

Our time horizons were at most those of a year in the Old Stone Age. In today's world, learning is much extended, and lifelong learning must be fostered by instilling the habits of learning and, moreover, learning how to learn, early on.


ARTICLE 3. WHETHER MATH NOT LEARNED NOW CAN BE LEARNED LATER.

It would seem that self-interested individuals can pick up whatever mathematics they come to realize they need at any time.

Objection 1. There are such things as critical periods for learning.

Objection 2. Businesses will not provide training, since trained workers can move elsewhere and take with them the training a firm has provided.

Objection 3. Later in life, workers have too many other objectives to accomplish, while kids have time on their hands. Furthermore, the brain is more supple at earlier ages.

Objection 4. Workers have short planning horizons set in the Environment of Evolutionary Adaptation (EEA), generally the Lower Paleolithic.

On the contrary, the Panel should investigate the genuine barriers to adult education and the extent to which mathematics education should be directed toward enable adults to learn math later, or "learning how to learn."

Reply to Objection 1. This may very well be the case, but none of the articles in the Journal for Research in Mathematics Education that I spotted go into the matter.

Reply to Objection 2. This is too general a problem and looks like rent-seeking on the part of businesses to get the taxpayer to foot the bill for training.

Reply to Objection 3. This could merely mean that further education is not all that it is cracked up to be.

Reply to Objection 4. This again is too general, as witness what is supposedly "too low" as savings rate (never mind that most investment comes from retained earnings by businesses), and says nothing about how big this molehill is.


ARTICLE 4. WHETHER THE NEED FOR MATHEMATICIANS CAN BE KNOWN?

It would seem that no one can say how many mathematicians there "ought" to be, since we can't even count them. There were, for example,  between 4 and 15 million scientists and engineers in 2003, depending on how they are counted (). International data is even less reliable. Such projections as do get made do little more than draw straight lines on logarithmic paper.

Objection 1. We very well know that mathematics, whether at the level of basic numeracy to that of pure mathematicians, is going to become so much more needed as computerization of basic work through the ability to make sophisticated new products as product cycles continue to shrink that it is pointless to demand quantification. School reform will lag so far behind the trends toward computerization and global competition that there is no chance that there will be too much mathematics taught in schools. This is what Mr. Jefferson called "the common sense of the matter." 

On the contrary, the Panel should strive to find a proper balance between requiring certain courses for all and making others available.

Reply to Objection 1. It is not at all clear that far too much math is required in schools already. Furthermore, courses are indeed available for those who want to further their mathematical learning.

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QUESTION 3. TRUTH TO BE TOLD TO A BENEVOLENT DESPOT (two articles)

ARTICLE 1. WHETHER THE PANEL BASICALLY WANTS TO TELL A BENEVOLENT DESPOT WHAT TO DO?

It would seem that there is a inbuilt bias toward saying "this is how I want the world to be" and then advising a benevolent despot about what to do. This "truth model," as James M. Buchanan calls it is entirely different from his "exchange model," which says that voters simply have different desires about what public goods they want provided, whence the basic problem become how to design a constitution so that voters get what they want for themselves without having to pay for too many things they don't want.

Objection 1. There is such a consensus about what students should learn in mathematics that there is really no difference between the truth and exchange models. The only real differences are over how best to achieve these aims, and finding out is the principle task of the Panel.

On the contrary, there are serious divisions about the aims of education, and they play into the culture wars. This war, according to James Davidson Hunter, The Culture Wars, is all about the existence of transcendental source of (absolute) morality vs. the contextualist (whom the absolutists call relativist) approach, which denies this. The Panel should strive to bring this conflict out in the open, for no one takes an extreme position on these matters.

Reply to Objection 1. This is just not true, as argued above, if only due to differences on what mathematics is good for, to the extent that this have even been thought about in the first place. Beyond this, there are four principle philosophies of education:

A. Perennialism, which urges the study of the classics, be it the Bible, the Koran, or the Little Book of Chairman Mao, whose principle task is that of moral education. Largely vanished from the public schools in America, a look at Ministry of Education websites in East Asia shows that specific time periods in Japan and Korea are set aside for moral education, specifically so named. Conservatives generally regard moral education as a good thing. This not specifically related to mathematics, however.

B. Essentialism. This, also called "Back to the Basics," holds that education should be organized around specific subjects and around the specifics of knowledge to be learned in each of these subjects. This approach also appeals to conservatives, as well as to expert panels who strive to draw up curriculum standards.

C. Progressivism. This approach envisions not so much a body of materials to be learned but rather the formation of habits of thought. (Dewey's concentration on training students to serve the common social good can be detached from this overall vision). This appeals to liberals.

D. Existentialism. This says that students should build their own course of study by following their various blisses. This also appeals to liberals, even if they characteristically are concerned with society-wide problems, and it assumes that young students both know their "particular circumstances of time and place" (Hayek, see below) about what paths to take to achieve whatever they want to achieve, regardless of how well they are prepared to cope in the world after they leave school. It  assumes that education is as much about the self-construction of personalities as anything else. It is the ultimate in free-market choice.


ARTICLE 2. WHETHER THE PANEL'S ADVICE IS ARBITRARY?

It would seem that experts are chosen by a Darwinian selection process. Those that deviate from the median by more than seven percent are deemed over the top, off the wall, and out to lunch. The consensus changes over time: one can join a panel today without insisting that trigonometry be mandated or even taught. Probably not so with geometry and certainly not so for anyone insisting that no math be required after junior high school. Any consensus will lag behind reality.

Objection 1. There is an objective, external world out there, and the process of deduction, induction, and abduction results in closer and closer approximation to this reality.

Objection 2. Cultural literacy does not require much knowledge of mathematics. Eric Donald Hirsch, a top expert in the subject, did not give Gödel's Incompleteness Theorem, certainly the most celebrated result in mathematics in the last century, among his 6,900 entries. See  for an online version of The New Dictionary of Cultural Literacy, third edition, 2002)

On the contrary, the Panel should think instead about what level of mathematical literacy can be achieved in popular culture as well as about what students should take in school and how the courses should be taught. General familiarity with statistics would benefit the citizens, as consumers and as voters alike, in helping them spot bogus arguments. This is hardly an arbitrary claim.

Reply to Objection 1. Whether or not it would be arbitrary to demand knowledge of this particular item, surely a broader appreciation  (his definition of mathematics is just "The study of numbers, equations, functions, and geometric shapes (see geometry) and their relationships. Some branches of mathematics are characterized by use of strict proofs based on axioms. Some of its major subdivisions are arithmetic, algebra, geometry, and calculus."

Reply to Objection 2. It would be remarkable if students should remember the quadratic equation! I have asked countless folks to recite it to me; only those who had majored in math remember it. (I actually had an occasion to use it, once, when I was fooling around with some data and came up with a quadratic equation.) The most that might be hoped for is that equations be presented, along with graphs,^ in popular culture, such as non-science television shows and pamphlets that get handed out on street corners. Yet I am reliably informed that even in Japan, where students score well on international math tests and who are driven hard by themselves, their parents, and their society, equations are absent in popular culture.

^(The first graph was drawn about 1340 by Nicole Orésme of the Universities of Paris and Oxford and was unknown to mathematicians of ancient Greece, Rome, China, and India. There may be examples of early graphs representing continuous change, but since this concept did not fit into their deep cultures, it was not developed. This is my favorite example of a second-nature notion that is so prevalent around the world that it seems like first nature.)

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QUESTION 4. THE STRUCTURE OF EDUCATIONAL GOVERNANCE

Objection 1. It would seem that the main issue is not what supposedly should be taught and how but why these reforms and strengthenings have not already been done. Teachers in America are so bound by bureaucratic rules that they cannot rely their own "the knowledge of the particular circumstances of time and place"^ and adopt their teaching accordingly. Liberate the teachers from the educrats!

^(The reference is to Friedrich Hayek's article, "The Use of Knowledge in Society," http://virtualschool.edu/mon/Economics/HayekUseOfKnowledge.html, which all Panelists are strongly urged to read.)

Objection 2. It would seem that research is directed too heavily toward "one size fits all," even as this is hotly denied. When NCLB gets reauthorized, care should be taken to allow experimentation and not punish trying out promising practices that eventually fail. A superior form of educational governance would view failures positively, as being necessary to learn from experience. Henry Petroski's engaging Success through Failure shows this for engineering, but it is applicable everywhere. Education reform is as much about setting up a learning network among educators as it is in achieving immediate results on standardized tests. The plain reality is that humans communicate largely by stories, meaning that a teacher will pay the greatest attention to a fellow teacher that has gained his respect and less to empirical studies no matter how good.^

^(The Panelists are also strongly urged to read Paul H. Rubin's Darwinian Politics: The Evolutionary Origin of Freedom (Rutgers UP, 2002). Paul is a professor of economics and law at Emory University and is well-versed both in Public Choice economics and socio-biology, whose respective paradigms of utility and fitness maximization conflict with each other. On page 177, he recounts the case of Ford Motor Company using statistical analysis to defend itself in the Pinto liability case, as deliberately including a dangerous feature in its design of the Pinto on grounds of its over-all cost-effectiveness, as the law indeed explicitly allowed. The prosecutors paraded the injured in front of the jury, and the jurors awarded huge damages to the injured.)

Objection 3. The largest (though unintended) effect of NCLB is to take control from teachers, schools, districts, and counties and concentrate them in the States. By mandating State-wide curriculum standards, any previous drift toward increasing critical thinking in the school curriculum, has been halted.

Objection 4. It is "thinking outside the box" that is more needed than simply feeding back answers on tests. No National Panel can possibly reach any consensus on what such "lateral thinking" consists of, to say nothing about how to foster its development. The only way to foster lateral thinking is to let teaching innovations bubble up from the bottom, even at the expense of failing to make Adequate Yearly Progress in some instances.

On the contrary, the Panel should pay the greatest attention to the structure of educational governance, along with thinking about what mathematics is good for and how better to teach it. How much within-State variation should be allowed is something for the Panel to dwell upon and about which to make representations to the reauthorizers of NCLB. Every mathematician (and economist) knows that It is rarely the case that optimum = maximum (which would lead to irresponsibility ) or optimum = minimum (which would stifle innovation). Indeed, establishing a learning network about successful and unsuccessful innovations could well lead to better (though not immediately measurable)  improvements in math education than any implementing of what are now regarded as better methods of teaching.

Reply to Objection 1. This would lead to irresponsibility. The choices are 1) choice (free market), 2) irresponsibility, and 3) accountability. There being no real prospect of privatizing education, the No Child Left Behind Act strengthens accountability, and strengthens it beyond what the States are capable of.

Reply to Objection 2. Such a learning network can indeed be set up, but it should still be up to the States to try only those reforms that ensure that the basics still be learned and that Adequate Yearly Progress continue to be made.

Reply to Objection 3. There is nothing that precludes changes in NCLB, when it is reauthorized, to allow different standards varying by school. Students at some schools could be assessed partly on the basis of better and better mastery of higher-level thinking skills. There is no need for this to come at the expense of failing to improve on the mastery of basic skills.

Reply to Objection 4. It will be well enough for States to define and measure these higher-level skills (which need to be applied only to certain schools or selected students within those schools.) If a learning network, that reaches across the States can be set up, more and more States can join in as they themselves see fit. Thought should be given to flexibility within counties, districts, and individual schools, but within the overall framework of making Adequate Yearly Progress according to State-wide standards that apply to all schools. It is not clear that there are genuine trade-offs to be made.


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QUESTION 5. TREATMENT OF THE GIFTED (two articles)

ARTICLE 1: WHETHER IT IS ASSUMED BY DEFAULT THAT ALL CHILDREN ARE GIFTED?

It would seem that the Panel members, all being gifted themselves, design curricular practices that work mostly for the gifted and pass over the heads of normal kids.

Objection 1. The Panel members have all been careful to realize this problem.

Objection 2. As an example of correcting this bias, the "new math" axiomatic approach has largely been abandoned, for introducing concepts too early, though it lives on math instruction. I was a new math guinea pig in 1959-60 but find that post-new math students manage to know, for example, what the intersection of sets are and what the distributive law states. It's just that these "New Math" ideas are introduced only later and are not subject to axiomatic treatment. Yet the much more recent "constructivist" approach to mathematics (also called the problem solving approach) has been subject to the same criticism as being inappropriately advanced conceptually for most students.

On the contrary, the Panel should scrutinize all studies they have for differential effects on different students of various programs, search the universe and its attics for other studies, and that future studies pay attention to this issue. They should bear in mind what the great sociologist of science, Robert King Merton, dubbed the Matthew effect, viz.:

For vnto euery one that hath shall be giuen,
and he shall haue abundance:
but from him that that not
shal be taken away
euen that which he hath.

--Matthew 25:29 (original 1611 spelling) (Parable of the Talents)

The Panel should be acutely aware of the intrusion of the culture wars into the writing of articles and their evaluation. The notion of a transcendental and absolute source for morality that dominates on one side (what was in the 1950s called the "squares," as opposed to the "mods") manifests itself psychologically in standing firm and not caving in. Both sides accuse the other side of caving in with great regularity . The [Henry] Petroskian virtue of "success through failure" is more needed than ever before. (This is also called "openness to experience" and is among the "Big Five" Personality Factors, clusters determined through factor analysis, the others being conscientiousness, agreeableness, extroversion, and neuroticism).

Reply to Objection 1. The bias toward assuming everyone is like oneself is so powerful that it creeps in despite the best intentions.

Reply to Objection 2. There are arguments that the problem-solving approach (that is, the pedagogy of presenting real-world problems to students rather than drilling them on formulae, whereby they construct their own understanding of mathematics on the fly) works at least as well as more traditional back-to-the-basics approach. See, Alan H. Schoenfeld, "Problem Solving in The United States, 1970-2007: Research and Theory, Practice and Politics" (Draft H, October 14, 2006.  To appear in: G. Törner, A. H. Schoenfeld, & K. Reiss (Eds.). Problem Solving Around the World--Summing up the State of the Art. Special issue of the Zentralblatt für Didaktik der Mathematik/International Reviews on Mathematics Education: Issue 1, 2008 (which I can supply).


ARTICLE 2: WHETHER THE SPECIAL NEEDS OF THE GIFTED ARE BEING IGNORED?

It would seem that the gifted are basically no different from the rest of the population and that they will flourish in any atmosphere.

Objection 1:  Penny Van Deur's study, "Gifted Reasoning and Advanced Intelligence," from the Australian Association for the Education of the Gifted and Talented, of which I can supply a copy,^ , argues that gifted children are able to negotiate and construct meta-mental maps, that is several diverse ways of approaching problems and, moreover begin to do so at the earliest ages.


^The essay was at , but many or most of its files have been moved to . Lots of articles on the gifted are still there.

Objection 2: Gifted children commonly get bored with school and even drop out. They do not achieve their potential.

Reply to Objection 1: The opportunity costs of specially catering to the gifted, as argued in Mara Sapon-Shevin's  Playing Favorites: Gifted Education and the Disruption of Community should not be slighted.

Reply to Objection 2. Gifted children, in fact, are better off in mainstream classrooms: "Many gifted programs, for example, focus on counseling able students or developing their social skills through activities such as leadership training and small-group interaction (e.g., Parker, 1983). In the name of improving gifted students' creativity, many programs forego substantial academic content and, instead, teach problem-solving skills in isolation from any particular academic content. These 'skills' are easily acquired and applicable only to narrowly-structured problems; they are, in consequence, of doubtful merit (McPeck, 1981). As Borland (1989, p. 174) notes, special instruction for the gifted often consists of 'an array of faddish, meaningless trivia--kits, games, mechanical step-by-step problem-solving methods, pseudoscience, and pop psychology.' Moreover, educators frequently dissuade students from attempting intellectually challenging programs by exaggerating the emotional and social risks of strategies like acceleration and early college attendance (Daurio, 1979)." From Aimee Howley, Edwina D. Pendarvis, and Craig B. Howley, "Anti-intellectualism in U.S. Schools, Educational Policy Analysis 1(6) (1994). 

On the contrary, it is cru

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QUESTION 6. THE PANEL AS A SHAM

ARTICLE 1. WHETHER THE PANEL IS A SHAM?

It would seem that the Panel is basically a sham. No real new research will be undertaken, any more than the Institute for Education Sciences has come up with substantial research in the several years of its existence. Reform is going to take place. Absent research, it will take the tried and true path of increasing test scores in line with conservative ideology (essentialism mostly) of drill, drill, drill, discipline, discipline. It is risky to do actual research, which might threaten the entrenched positions of ideologues. (This is all but argued by Edward A Silver: "Improving education research: Ideology or science?" Journal for Research in Mathematics Education, 34(2) (2003 March) ,  p. 106f, of which a copy can be furnished.)

Objection 1. These charges are so predictable that they will be hurled regardless of the facts of the situation and therefore should be ignored. The Panel members do indeed represent a wide variety of points of views.

On the contrary, the Panel should address the matter of the culture wars up front and relate them to various philosophies of mathematics education.

Reply to Objection 1. The culture wars are nevertheless real.

Reply to Objection 2. In an ideal world, these relationships would be better known, but in any case doing well on tests is important for morale, and doing well encourages students and citizens alike to  to continue to strive.

ARTICLE 2. WHETHER SCORING WELL ON TESTS IS AN END IN ITSELF?

It would seem that the Panel is a sham, for the entire business of raising scores on tests, relevant or not as an indicator of becoming economically "competitive," is less important than winning this symbolic competition. 

Objection 1. Good preparation in mathematics is increasingly important in a world where production is becoming more and more based upon applying science and using engineering skills.

Objection 2. The tests we have are good measures of the skills that will be more and more needed in the future economy.

Objection 3. Being better prepared in mathematics will enable American workers to do better in international economic competition.

On the contrary, while scoring well on tests is not without its symbolic value, and even if test scores are imperfect indicators, having indicators is indispensable. Those who rail against them are nevertheless quite willing to use them in support of their ideas.

Reply to Objection 1. However true this is, and still only a small number of workers will be engaged in jobs that actually utilize mathematics beyond arithmetic, wee know from biology that animals engage in ritualized combat, that when beta-male challenges alpha-male the winner does not kill the loser but accepts a ritual sign of submission. In human warfare, representatives from two parties can be chosen to engage in one-to-one combat rather than the winning side exterminating the losing side. I append a familiar version.

Reply to Objection 2. Since the relationship between mathematics education and national "competitiveness" is nearly unknown, and since "competitiveness" has no operational definition anyhow, except GDP/capita (just like "access" to education winds up getting measured by enrollment), it is well enough that U.S. students score high on these tests. For the same reason, the Iron Curtain countries thought it so important that they win in the get a large number of medals in the Olympic games that they cheated. They thought it tremendously important that their very best athletes run a fraction of a second faster than other countries' best athletes, even though this says next to nothing about the average speed of the members of these countries, since the distribution of running speeds is not normal at the extreme ends.

Reply to Objection 3. Spokesmen for education in countries in the Far East, such as Japan, China, and Singapore regularly complain that, while their students do very well on math tests, they cannot think, that is think creatively. There aren't any really good tests of independent thinking, and no one know how to foster it.

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QUESTION 7: TABOO ISSUES

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APPENDIX 1: CHARTER OF THE NATIONAL MATHEMATICS ADVISORY PANEL

http://www.ed.gov/about/bdscomm/list/mathpanel/charter.pdf

Authority 

The National Mathematics Advisory Panel (Panel) is established within the Department of Education under Executive Order 13398 by the President of the United States and governed by the provisions of the Federal Advisory Committee Act (FACA) (P.L. 92463, as amended; 5 U.S.C. App.). 

Background 

In order to keep America competitive, support American talent and creativity, encourage innovation throughout the American economy, and help State, local, territorial, and tribal governments give the Nation's children and youth the education they need to succeed, it shall be the policy of the United States to foster greater knowledge of and improved performance in mathematics among American students. 

Purpose and Functions 

The Panel shall advise the President and the Secretary of Education (Secretary) on the conduct, evaluation, and effective use of the results of research relating to proven-effective and evidence-based mathematics instruction, consistent with policy set forth in section 1 of the Executive Order. In carrying out its mission, he Panel shall submit to tthe President, through the Secretary, a preliminary report not later than January 31, 2007, and a final report not later than February 28, 2008. 

The Panel shall obtain information and advice as appropriate in the course of its work from: 

1. Officers or employees of Federal agencies, unless otherwise directed by the head of the agency concerned; 
2. State, local, territorial, and tribal officials; 
3. Experts on matters relating to the policy set forth in section 1; 
4. Parents and teachers; and 
5. Such other individuals as the Panel deems appropriate or as the Secretary may direct. 

Structure 

The Panel shall consist of no more than 30 members as follows: 

1. No more than 20 members from among individuals not employed by the Federal Government, appointed by the Secretary for such terms as the Secretary may specify at the time of appointment; and 
2. No more than 10 members from among officers and employees of Federal agencies, designated by the Secretary after consultation with the heads of the agencies concerned. The Secretary shall designate a Chair of the Panel from among the group of 20 members who are not employed by the Federal Government. Non-Federal members of the Panel shall serve as Special Government Employees (SGEs). As SGEs, the members will provide personal and independent advice based on their own individual expertise and experience. 

Meetings 

Subject to the direction of the Secretary, the Chair, in consultation with the Designated Federal Official (DFO), shall convene and preside at meetings of the Panel, determine its agenda, direct its work, and, as appropriate, deal with particular subject matters, and establish and direct the work of subgroups of the Panel that shall consist exclusively of members of the Panel. 

The Secretary or her designee shall name the Designated Federal Official (DFO) to the Panel. The Panel shall meet at the call of the DFO or the DFO's designee, and this person shall be present for all meetings. The DFO will work in conjunction with the Chair to convene meetings of the Panel. 
Meetings are open to the public except as may be determined otherwise by the Secretary in accordance with Section 10(d) of the FACA. Adequate public notification will be given in advance of each meeting. Meetings are conducted and records of the proceedings kept as required by applicable laws. A majority of the members of the Panel shall constitute a quorum but a lesser number may hold hearings. 

Estimated Annual Cost 

Members of the Panel who are not officers or employees of the United States shall serve without compensation and may receive travel expenses, including per diem in lieu of subsistence, as authorized by law for persons serving intermittently in Government service (5 U.S.C. 5701-5707), consistent with the availability of funds. 

Funds will be provided by the Department of Education to administer the Panel. The estimated annual person-years of staff support are four (4) Full-Time Equivalents. The estimated two-fiscal-year cost will be approximately $1,000,000. 

Report 

The Panel shall submit to the President, through the Secretary, a preliminary report not later than January 31, 2007, and a final report not later than February 28, 2008. Both reports shall, at a minimum, contain recommendations, based on the best available scientific evidence, on the following: 

1. The critical skills and skill progressions for students to acquire competence in algebra and readiness for higher levels of mathematics; 
2. The role and appropriate design of standards and assessment in promoting mathematical competence; 
3. The processes by which students of various abilities and backgrounds learn mathematics; 
4. Instructional practices, programs, and materials that are effective for improving mathematics learning; 
5. The training, selection, placement, and professional development of teachers of mathematics in order to enhance students' learning of mathematics; 
6. The role and appropriate design of systems for delivering instruction in mathematics that combine the different elements of learning processes, curricula, instruction, teacher training and support, and standards, assessments, and accountability; 
7. Needs for research in support of mathematics education; 
8. Ideas for strengthening capabilities to teach children and youth basic mathematics, geometry, algebra, and calculus and other mathematical disciplines; 
9. Such other matters relating to mathematics education as the Panel deems appropriate; and 
10. Such other matters relating to mathematics education as the Secretary may require.
 
The Secretary may require the Panel, in carrying out subsection 2(b) of Executive Order 13398, to submit such additional reports relating to the policy set forth in section 1 of the Executive Order. 

Termination 

Unless extended by the President, this Advisory Panel shall terminate April 18, 2008. 
This charter expires April 18, 2008. 
Approved: 


___________________________ 
Date Secretary 
Filing date:

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APPENDIX 2: I Samuel 17

1 Now the Philistines gathered together their armies to battle, and were gathered together at Shochoh, which belongeth to Judah, and pitched between Shochoh and Azekah, in Ephes-dammim.
2 And Saul and the men of Israel were gathered together, and pitched by the valley of Elah, and set the battle in array against the Philistines.
3 And the Philistines stood on a mountain on the one side, and Israel stood on a mountain on the other side: and there was a valley between them.
4 And there went out a champion out of the camp of the Philistines, named Goliath , of Gath, whose height was six cubits and a span.
5 And he had an helmet of brass upon his head, and he was armed with a coat of mail; and the weight of the coat was five thousand shekels of brass.
6 And he had greaves of brass upon his legs, and a target of brass between his shoulders.
7 And the staff of his spear was like a weaver's beam; and his spear's head weighed six hundred shekels of iron: and one bearing a shield went before him.
8 And he stood and cried unto the armies of Israel, and said unto them, Why are ye come out to set your battle in array? am not I a Philistine, and ye servants to Saul? choose you a man for you, and let him come down to me.
9 If he be able to fight with me, and to kill me, then will we be your servants: but if I prevail against him, and kill him, then shall ye be our servants, and serve us.
10 And the Philistine said, I defy the armies of Israel this day; give me a man, that we may fight together.
11 When Saul and all Israel heard those words of the Philistine, they were dismayed, and greatly afraid.
12 Now David was the son of that Ephrathite of Bethlehem Judah, whose name was Jesse; and he had eight sons: and the man went among men for an old man in the days of Saul.
13 And the three eldest sons of Jesse went and followed Saul to the battle: and the names of his three sons that went to the battle were Eliab the firstborn, and next unto him Abinadab, and the third Shammah.
14 And David was the youngest: and the three eldest followed Saul.
15 But David went and returned from Saul to feed his father's sheep at Bethlehem.
16 And the Philistine drew near morning and evening, and presented himself forty days.
17 And Jesse said unto David his son, Take now for thy brethren an ephah of this parched corn, and these ten loaves, and run to the camp to thy brethren;
18 And carry these ten cheeses unto the captain of their thousand, and look how thy brethren fare, and take their pledge.
19 Now Saul, and they, and all the men of Israel, were in the valley of Elah, fighting with the Philistines.
20 And David rose up early in the morning, and left the sheep with a keeper, and took, and went, as Jesse had commanded him; and he came to the trench, as the host was going forth to the fight, and shouted for the battle.
21 For Israel and the Philistines had put the battle in array, army against army.
22 And David left his carriage in the hand of the keeper of the carriage, and ran into the army, and came and saluted his brethren.
23 And as he talked with them, behold, there came up the champion, the Philistine of Gath, Goliath by name, out of the armies of the Philistines, and spake according to the same words: and David heard them.
24 And all the men of Israel, when they saw the man, fled from him, and were sore afraid.
25 And the men of Israel said, Have ye seen this man that is come up? surely to defy Israel is he come up: and it shall be, that the man who killeth him, the king will enrich him with great riches, and will give him his daughter, and make his father's house free in Israel.
26 And David spake to the men that stood by him, saying, What shall be done to the man that killeth this Philistine, and taketh away the reproach from Israel? for who is this uncircumcised Philistine, that he should defy the armies of the living God?
27 And the people answered him after this manner, saying, So shall it be done to the man that killeth him.
28 And Eliab his eldest brother heard when he spake unto the men; and Eliab's anger was kindled against David, and he said, Why camest thou down hither? and with whom hast thou left those few sheep in the wilderness? I know thy pride, and the naughtiness of thine heart; for thou art come down that thou mightest see the battle.
29 And David said, What have I now done? Is there not a cause?
30 And he turned from him toward another, and spake after the same manner: and the people answered him again after the former manner.
31 And when the words were heard which David spake, they rehearsed them before Saul: and he sent for him.
32 And David said to Saul, Let no man's heart fail because of him; thy servant will go and fight with this Philistine.
33 And Saul said to David, Thou art not able to go against this Philistine to fight with him: for thou art but a youth, and he a man of war from his youth.
34 And David said unto Saul, Thy servant kept his father's sheep, and there came a lion, and a bear, and took a lamb out of the flock:
35 And I went out after him, and smote him, and delivered it out of his mouth: and when he arose against me, I caught him by his beard, and smote him, and slew him.
36 Thy servant slew both the lion and the bear: and this uncircumcised Philistine shall be as one of them, seeing he hath defied the armies of the living God.
37 David said moreover, The Lord that delivered me out of the paw of the lion, and out of the paw of the bear, he will deliver me out of the hand of this Philistine. And Saul said unto David, Go, and the Lord be with thee.
38 And Saul armed David with his armour, and he put an helmet of brass upon his head; also he armed him with a coat of mail.
39 And David girded his sword upon his armour, and he assayed to go; for he had not proved it. And David said unto Saul, I cannot go with these; for I have not proved them. And David put them off him.
40 And he took his staff in his hand, and chose him five smooth stones out of the brook, and put them in a shepherd's bag which he had, even in a scrip; and his sling was in his hand: and he drew near to the Philistine.
41 And the Philistine came on and drew near unto David; and the man that bare the shield went before him.
42 And when the Philistine looked about, and saw David, he disdained him: for he was but a youth, and ruddy, and of a fair countenance.
43 And the Philistine said unto David, Am I a dog, that thou comest to me with staves? And the Philistine cursed David by his gods.
44 And the Philistine said to David, Come to me, and I will give thy flesh unto the fowls of the air, and to the beasts of the field.
45 Then said David to the Philistine, Thou comest to me with a sword, and with a spear, and with a shield: but I come to thee in the name of the Lord of hosts, the God of the armies of Israel, whom thou hast defied.
46 This day will the Lord deliver thee into mine hand; and I will smite thee, and take thine head from thee; and I will give the carcases of the host of the Philistines this day unto the fowls of the air, and to the wild beasts of the earth; that all the earth may know that there is a God in Israel.
47 And all this assembly shall know that the Lord saveth not with sword and spear: for the battle is the Lord's, and he will give you into our hands.
48 And it came to pass, when the Philistine arose, and came and drew nigh to meet David, that David hasted, and ran toward the army to meet the Philistine.
49 And David put his hand in his bag, and took thence a stone, and slang it, and smote the Philistine in his forehead, that the stone sunk into his forehead; and he fell upon his face to the earth.
50 So David prevailed over the Philistine with a sling and with a stone, and smote the Philistine, and slew him; but there was no sword in the hand of David.
51 Therefore David ran, and stood upon the Philistine, and took his sword, and drew it out of the sheath thereof, and slew him, and cut off his head therewith. And when the Philistines saw their champion was dead, they fled.
52 And the men of Israel and of Judah arose, and shouted, and pursued the Philistines, until thou come to the valley, and to the gates of Ekron. And the wounded of the Philistines fell down by the way to Shaaraim, even unto Gath, and unto Ekron.
53 And the children of Israel returned from chasing after the Philistines, and they spoiled their tents.
54 And David took the head of the Philistine, and brought it to Jerusalem; but he put his armour in his tent.
55 And when Saul saw David go forth against the Philistine, he said unto Abner, the captain of the host, Abner, whose son is this youth? And Abner said, As thy soul liveth, O king, I cannot tell.
56 And the king said, Inquire thou whose son the stripling is.
57 And as David returned from the slaughter of the Philistine, Abner took him, and brought him before Saul with the head of the Philistine in his hand.
58 And Saul said to him, Whose son art thou, thou young man? And David answered, I am the son of thy servant Jesse the Bethlehemite.

It should be noted that there are contrary happenings, recorded in the same book:

1 Samuel 25:22
So and more also do God unto the enemies of David, if I leave of all that pertain to him by the morning light any that pisseth against the wall.

1 Samuel 25:34
For in very deed, as the Lord God of Israel liveth, which hath kept me back from hurting thee, except thou hadst hasted and come to meet me, surely there had not been left unto Nabal by the morning light any that pisseth against the wall.

1 Kings 14:10
Therefore, behold, I will bring evil upon the house of Jeroboam, and will cut off from Jeroboam him that pisseth against the wall, and him that is shut up and left in Israel, and will take away the remnant of the house of Jeroboam, as a man taketh away dung, till it be all gone.

1 Kings 16:11
And it came to pass, when he began to reign, as soon as he sat on his throne, that he slew all the house of Baasha: he left him not one that pisseth against a wall, neither of his kinsfolks, nor of his friends.

1 Kings 21:21
Behold, I will bring evil upon thee, and will take away thy posterity, and will cut off from Ahab him that pisseth against the wall, and him that is shut up and left in Israel.

2 Kings 9:8

For the whole house of Ahab shall perish: and I will cut off from Ahab him that pisseth against the wall, and him that is shut up and left in Israel.

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THE DUCHESS'S EPILOGUE

When all was over--that is, when it became clear that nothing important is ever over in mathematics education--in came the Duchess and drew her morals.

Anthropological Moral 1: Our time horizons were set during the Old Stone Age.

Anthropological Moral 2: Contests are highly ritualized, in animal and human societies alike. Beware of elevating scoring well on tests to an end in itself.

Cultural Moral: Bring the Culture Wars out in the open. There will be less dissension and more openness to pick the best from both sides. Polarization is only apparent.

Economics Moral 1: The forces of supply and demand will solve most problems. If they do not, consider whether the cure is worth its cost.

Economics Moral 2: Beware of using terms like competitiveness that have no operational meaning. What would happen if the United States ceased to be "competitive"?

Economic Moral 3: Always consider the cost of lost opportunities, such as the neglect of education for the gifted.

Epistemological Moral: Do not fall back on ideological beliefs where knowledge is not firm. Rather, sponsor further research.

Ethical Moral: Do not just say "this is how I want the world to be."

Metaphysical Moral: Learning comes from within; it does not come from without. "Lead a horse to water, but you can't make him drink." To be sure, rare teachers can inspire a willingness to learn, but in practice only 10-15 percent of the variation in learning comes from variation in methods of course instruction, less if learning is learning habits of thought, rather than just doing well on tests. Free will is hard to measure, even to proxy, so it is rarely considered in research in mathematics education.

Pedagogical Moral: How mathematics education is transferred to other realms is more important that the math itself, since few students will ever use math beyond what they learn in junior high school.

Political Moral: The structure of educational governance is the overriding one. Always ask why reforms have not been implemented. What are the institutional barriers?

Finally, the moral of this set of moral is, Check your premises.

[This was written and prepared entirely at home. It was not part of my work. Not one reader uncovered the parody.]