Big Surprise!

One of the feature articles in the January 2005 issue of Scientific American states...

Boosting people's sense of self-worth has become a national preoccupation. Yet surprisingly, research shows that such efforts are of little value in fostering academic progress.

"Surprisingly"? I don't see what's so surprising. I'm not surprised. Are you surprised? The entire article is worth a read and can be found here.

Teachers Who Fail

http://www.heraldtribune.com/apps/pbcs.dll/article?AID=/20041212/NEWS/412120357/

More than half a million Florida students sat in classrooms last year in front of teachers who failed the state's basic skills tests for teachers. Many of those students got teachers who struggled to solve high school math problems or whose English skills were so poor they flunked reading tests designed to measure the very same skills students must master before they can graduate.

A Herald-Tribune investigation has found that fully a third of teachers, teachers' aides and substitutes failed their certification tests at least once. The Herald-Tribune found teachers who had failed in nearly every school in each of the state's 67 counties.

Nine percent of teachers failed portions of the tests at least four times, according to the Herald-Tribune study.

[Thanks to DVD for the link.]

Apologies

Apologies for my lack of recent posting in this blog. I am by no means abandoning this blog. It's just that, as I mentioned before, it is difficult to come up with material suitable for this format. I am continuing to work on an analysis of the NCTM Standards and hope to have some posts on this topic which will be of manageable length for this blog. I am also reading through the new (old?) California curriculum and expect to be able to compose a short review in the near future. Beyond that, I'm not sure what to do.

Comments, questions, suggestions or rants are always welcome.

Quick Blurb

High school students in Hong Kong, Finland and South Korea do best in mathematics among those in 40 surveyed countries while students in the United States finished 28th.

The survey also questioned students about their own views of themselves and their work, and found that while good students were more likely to think they were good, countries that did well often had a large number of students who did not feel they were doing well. In the United States, 36% of the students agreed with the statement, "I am just not good at mathematics," while in Hong Kong, 57% agreed. In South Korea the figure was 62%. Of the United States students, 72% said they got good grades in mathematics, more than in any other country. In Hong Kong, only 25% of the students said they got good marks, the lowest of any country.

[Thanks to DVD for providing the NYT link.]

National Test of Student Math Skills

http://www.cnn.com/2004/EDUCATION/11/18/math.test.ap/index.html

The national test of student math skills is filled with easy questions, raising doubts about recent gains in achievement tests, a study contends. On the eighth-grade version of the test, almost 40% of the questions address skills taught in first or second grade, according to the report by Tom Loveless, director of the Brown Center on Education Policy at The Brookings Institution.

So, is the test flawed? Maybe...

The study analyzed questions from the 2003 math tests, and then determined a grade level for those questions based on the Singapore math textbook program. Loveless said he chose that program because of its clarity and strong international reputation, and he said it compared well to the math-class sequences used in states such as California and North Carolina. But using Singapore as a model presents skewed results, said Sharif Shakrani, deputy executive director of the National Assessment Governing Board. Math is taught differently in that country, with heavy concentration on computation early before other topics are introduced. U.S. schools go for breadth, he said, with more math skills to cover each year. Overall, he said, the questions on the national in fourth grade and eighth grade are commensurate with what's being taught in those grades.

I think I understand. If Mr. Shakrani is to be believed, it's not the TEST that's flawed; it's the instruction. Math is taught "differently" (meaning "well") in Singapore...

"I contend that if we do what he suggests, moving to much more complex skills, it would be akin to giving a test in Russian," Shakrani said. "We already are not doing well. If you increase the cognitive function of the math concepts and the way you test them, you will end up with scores so low you will not be able to make sense of the results."

OK, I guess I don't understand after all. WTH does this mean? Don't test at the appropriate level because the scores would be "too" low?? Then what are the exams measuring . . . and what are they supposed to measure???

Number of foreign graduate students down

A new survey indicates the number of foreign graduate students enrolling for the first time at American universities is down 6 percent this year -- the third straight decline after a decade of growth. Educators worry the trend is eroding America's position as the world's leader in higher education.

But the results of the survey, of 122 member institutions by the Council of Graduate Schools, are still alarming to educators. American universities are highly dependent on foreign students for teaching and research help, particularly in the sciences and in engineering, a field in which foreigners comprise 50 percent of graduate enrollment.

How did we allow ourselves to get into a situation where if foreign students do not arrive at our graduate schools in sufficient numbers, these institutions can't function properly ("alarming" is the word used in the article)? Surely, policies should be enacted to ensure that we have sufficient home-grown talent to fill these slots?

California Mathematics Draft Framework

Wow! I’m genuinely impressed. California, which so often is a force for idiocy in education, is blazing trails towards a new (old?) mathematics curriculum.

http://www.cde.ca.gov/ci/ma/cf/index.asp

I’ll go over all the standards in more detail later, but for now I’ll make a few quick observations on the chapter on geometry.

• Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
• Students construct, and judge the validity of, a logical argument and give counterexamples to disprove a statement.
• Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
• Students write geometric proofs, including proofs by contradiction.
• Students prove basic theorems involving congruence and similarity.
• Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
• Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
• Students prove theorems and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
• Students prove the Pythagorean theorem.
• Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle.
• Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
• Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.

Sincere kudos to the drafters of this curriculum!

Review of "A Core Curriculum"

Some comments on "A Core Curriculum"

Page 1: "A need to enlarge the scope of this Newtonian mathematics curriculum began to emerge as mathematics increasingly became a tool in the social sciences."

What exactly do social scientists need from math that is not covered by the standard curriculum? A course in statistics in lieu of pre-calculus or calculus has always been available for students so inclined. Beyond that, the traditional curriculum provides the training to "think mathematically" of which NCTMers are so enamoured. Replacing the "Newtonian" mathematics curriculum with a bunch of vaguely connected statistical concepts dooms students to a lifetime of misunderstanding and misapplying statistics to their daily lives. The unassailable fact is that calculus is still the keystone to understanding all of this allegedly "new" math.

Page 8: Dart Throwing Exercise - "Measure a distance eight feet from the target and place a piece of tape on the floor. Standing behind the tape, the dart thrower throws some number of times at the target."

The purpose of this exercise is basically to do a simulation to discover the percentage of darts that fall within the white area on the target. The mathematical point to take away is that the white area is 78.54% of the square regardless of how many circles are circumscribed in the square. Of course, the number of darts that could reasonably be thrown by the students is not even remotely sufficient to get a reasonable accurate approximation. Even if sufficient accuracy were somehow obtained, statistical fluctuations will ensure that the percentages will never be the same for all the scenarios.

Page 34: "Assessment matters: One approach is to incorporate more student self-assessment. Ask students to write a brief self-assessment after they have completed a written assignment. Writing in a journal is also a good way to get students to reflect on their own performance."

Page 37: "When the instructional emphasis is on concept building through situations reflecting real-world questions and activities, the assessment should be of a similar nature. Open-ended holistically scored questions, interviews, observation of group work, testing with the use of physical models like those used in instruction, and student self-assessment are appropriate approaches."

I will let these monuments of edu-speak stand on their own.

Page 73: "Our axiom is that concepts are more powerful than procedures and more accessible to more students."

Brilliant. Evidence? Experiments? Research? Fuggedaboutit. Assume the very statement you're trying to prove. Then nobody can challenge you. You'd think mathematics educators would be familiar with the meaning of "axiom," but let's go through the motions. "A self-evident or universally recognized truth; A self-evident principle or one that is accepted as true without proof as the basis for argument." Should we really be accepting the statement above as an "axiom"? Come on now.

Page 113: "Old habits inconsistent with the new must be discarded."

Whether or not the old habits were effective and whether or not the new habits are effective.

Page 115: "The half-life of the education of an engineer has been estimated at ten years. In one decade, half of an engineer's training will become obsolete."

This statement is constantly used in support of the plan to replace "Newtonian" mathematics with a statistics-based curriculum. To me, this statement provides support for the diametrically opposite position. If engineers trained with "Newtonian" mathematics can survive the obsolescence of half their knowledge base and continue to function effectively, then this is precisely the training we should give everyone.

Page 117: "Use inductive reasoning to develop ideas where deduction requires too many underpinnings. Consider postulating important chunks of content, then use deductive reasoning from that base of understanding. Conclude coursework with modest deductive systems."

This would destroy the entire logical structure of the deductive system under investigation. As the NCTM should understand, the purpose of introducing deductive systems in high school is not so much to present the material itself but rather to present the concept of a deductive system. Presenting it in the way suggested above would destroy the rationale for doing this and eliminate any educational benefit in doing it at all.

Page 117: "Encourage students to investigate questions of interest to themselves."

Without the teacher controlling this process, most of the questions investigated will be of no value and irrelevant to the content. The uncomfortable (to the ed school powers-that-be) fact is that certain avenues of investigation are fruitful and most others are not; the teacher must provide this direction.

"Reshaping Assessment"

"A Core Curriculum", NCTM (1992), page vii

Analysis of students' written work remains important. However, single-answer paper-and-pencil tests are often inadequate to assess the development of students' abilities to analyze and solve problems, make connections, reason mathematically, and communicate mathematically. Potentially richer sources of information include student-produced analyses of problem situations, solutions to problems, reports of investigations, and journal entries. Moreover, if calculator and computer technologies are now accepted as part of the environment in which students learn and do mathematics, these tools should also be available to students in most assessment situations.

Let's analyze this innocuous looking statement from the preface of A Core Curriculum.

"Single-answer paper-and-pencil tests are often inadequate." – Why? What's the evidence for this statement?

"Student-produced analyses of problem situations" – What does this mean?

"solutions to problems" – Exactly how is this different from a "single-answer paper-and-pencil test"?

"Reports of investigations" – What exactly is a high school student qualified to "investigate"? Certainly, the occasional research project may be helpful and even desireable, but for the most part traditional techniques are a much more efficient method for imparting mathematical knowledge than student investigations. Of course, the premise of NCTM math is that this statement is not true. Fine, what evidence supports the thesis that student investigations are more efficient? I've never seen any.

"journal entries" – too stupid for words.

Most interesting of all is the last sentence. "IF calculator and computer technologies are now accepted . . . these tools should be available." They are using the very statement for which they are attempting to provide justification as their hypothesis. Nice.

In summary, what evidence supports the counter-intuitive notions that (a) high school students can handle complex, multi-step, open-ended (often ill-posed) investigations and (b) this is the best way to teach basic mathematical concepts?

Scary

I had to read the following paragraph several times to make sure I was not misunderstanding what was said.

Dr. Martin Haberman, Distinguished Professor in the School of Education at the University of Wisconsin-Milwaukee, in a talk entitled Can Teacher Education Close the Achievement Gap? given to AERA on April 2, 2002, in New Orleans, said:

The most efficient ways of recruiting and selecting the wrong people at the initial teacher preparation level i.e. those who will never take positions teaching diverse children in poverty or who will quit or fail if they deign to try - are the criteria most commonly used: a composition on why I want to teach, G.P.A., letters of reference, a basic skills test, etc. These irrelevant criteria are frequently used in traditional and alternative certification programs. Actually, undergraduate GPA does predict. If it is extremely high in courses outside of education it predicts quitting and failure.

I urge you to read his entire speech!

Response to Comment

Superdestroyer wrote…

You may want to check the real statistics on graduate students in engineering in the US. Graduate students in engineering are more white than anything else, more native [than] foreign and most of the foreign went undergraduate in the US. However, the number have gone down a little.

Thanks for the link to this data. It was most enlightening. However, you seem to have completely missed the point of what this data is actually showing. I was talking about mathematics and physics, and you brought up engineering. If you look at the data you provided, you will see that engineering actually has a considerably LARGER percentage of temporary visa holders than either math or physics. In fact, among all the fields represented in your data, engineering is the one with the largest number as well as the largest increase in temporary visa holders, and the percentage in 2002 stood at an alarming 49%! This is exactly what I’m talking about. I’m not sure exactly what you’re debating, unless it's that 51% native - 49% foreign is not a problem.

Your claims that students are more "white than anything else" that "most of the foreign went undergraduate in the US" are absolutely NOT supported by this data.

Edit (9/16/2007): In attempting to create a new graph with more recent NSF data, I accidentally overwrote the graph that was part of this post. Note that on the new graph using data through 2005 the engineering foreign visa student percentage is above 50%. See my blog post dated 9/16/2007 for more details.

Also, probably the total number of high students taking calculus in the US is greater than it has ever been before. Why do you keep claiming that US schools are failing. How many US high students graduate in 1964 were taking calculus?

I checked; this is actually correct. The number of students taking calculus has gone up consistently since 1955 (the first year the AP Calculus exam was offered). Even more encouraging, the percent of students taking AP Calculus has increased. But these are absolutely NOT the students about whom I’m concerned. Even now, only about 2% of high school students take calculus. Anybody in this elite group is already a gifted math student who is going to do well regardless of what the school system inflicts on him or her. The students about whom I actually am concerned – and the ones whom US schools are indeed failing – are the ones who could be successful in math, science or engineering, but who are not provided with the skills they need because of the NCTM math idiocy pervading our high school mathematics curriculum.

A Non-Math Example

Just to clarify what I'm talking about, I found an example from outside the realm of math and science. This should allow me to further illustrate and expand my point without any confusion.

The following are the requirements for a social studies education major at a well-known and well-respected private school (which falls into the Top National University category in the USN&WR rankings). By the way, if anyone recognizes the school, I am in no way trying to pick on this particular institution; the social studies education major is pretty much the same across the country.

Subject matter:
GEOG 101 – Global Environment: Understanding Physical Geography
ECON 101 – Economic Principles and Problems
PSYC 101 – General Psychology
SOC 101 – Introduction to Sociology OR ANTH 101 – Introduction to Anthropology
POL 101 – American Government and Politics
GEOG 130 – Introduction to Human Geography
POL 150 – Comparative Government and Politics
HIST 201, 202 – World Civilization OR HIST 211, 212 – United States History
HIST 364 – History of [State]
One sophomore-level geography course
One sophomore-level economics course
One junior-level psychology course
One junior-level history course

Education:
Educ 276 – Exploration of Teaching
Educ 286 - Instructional Technology in Teaching
Educ 350 - Adolescent Development in an Education Context
Educ 352 - Exceptional Education
Educ 353 - Multicultural Education
Educ 377 - Teaching Methods and Instruction
Educ 378 - Practicum in Secondary Education
Educ 379 - Classroom Management
Educ 476 - Internship

What exactly is a graduate of this program qualified to teach? This "major" consists of courses in a bunch of different fields, many of them 101 survey-type courses with no more than 3 courses in any single field, with the exception of history which boasts 4. But of course, history is the field with the most appalling gap of all, since it doesn't even include introductions to both U.S. History and World History, subjects that all social studies teachers end up teaching at some point! So we are told that someone is qualified to teach a subject when they've never taken even a college survey course in it.

This major is clearly in need of toughening up. It matters not if this major is populated by or avoided by "upper middle class white kids" or "Asian kids." If this major is toughened up and students who would complete it in its current state are scared away after it becomes "too tough," then them's the breaks and that can't be helped. We can not allow candidate supply & demand considerations to affect curriculum decisions.

It has been my experience in discussing this with a wide range of people that this major's deficiencies are immediately obvious to most people. Unfortunately, the same thing can not be said about the deficiencies in college math education majors. Perhaps that is because most people are more comfortable with the subjects above than they are with mathematics. Over the course of the next couple of weeks, I will attempt to make a strong case that the mathematics education major (and by extension, the high school math curriculum) is as deficient as the major described above, even if this is less obvious at first glance.

Response to Comment

Superdestroyer wrote...

Your logic reminds me of the military logic of when they are short of good special ops troops, they always come up with the idea of making training harder. I do not see how making high school math harder is going to encourage more upper middle class white kids to go into it.

Your lack of logic (or lack of reading comprehension or interest in arguing for argument's sake -- I'm not really sure what's going on here) boggles my mind. You are completely missing my point; I can not believe that my words are that unclear. I will attempt to elaborate one more time, but I grow seriously weary of this.

1. I am NOT saying that we are short of native students who complete mathematics courses. I AM saying that students who complete the pre-calculus curriculum in US high schools are not able to compete in college math courses with students who complete the pre-calculus curriculum in foreign high schools. Further, students who complete an undergraduate math major in US colleges are not able to compete in graduate math courses with students who complete an undergraduate math major in foreign colleges. To extend your special ops analogy, if we had plenty of special ops troops but when we put them against the special ops troops of other countries, they got slaughtered, surely the proper response absolutely would be more training. Do you disagree?
2. I DON'T have the slightest interest in encouraging more upper middle class white kids to go into mathematics. I DO have an interest in getting (a) more students who complete US high school mathematics to have the tools to compete in an undergraduate math major and (b) more students who complete an undergraduate math major to have the tools to compete in a graduate math program. How you continue to translate this into "upper middle class white" is something I can not even begin to comprehend. Why upper-middle class? Presumably, mathematical ability is distributed normally with respect to income? I would assume that lower-middle class, upper class and lower class students can be as successful as upper-middle class students in math, science and engineering, so I don't understand your focus on that single group.
3. The addition of term "white" just confuses me. I have repeatedly attempted to make clear that my point is NOT a racial or ethnic or cultural one. My point IS about U.S. versus foreign training. U.S. doesn't mean "white"; it could just as well mean black, Asian, Native American, or Latino (not that this last one is actually a race, but I will bow to the common mis-use of the word). How anything that I've said can be construed as referring to "white" only is unclear to me. When I point to a list of names as an example, my point is NOT that the names are not "white", but rather that (based on my experience - more on this below) they are not U.S. citizens. Yes, I understand that the United States are very much a melting pot, but in my fairly extensive observation of this matter, a Lu or Kim or Nguyen or Yagamuchi in a graduate mathematics program is overwhelmingly more likely to be from a foreign academic background than a U.S. academic background.

It will just encourage more of them to go into law so that they can spend their time second guessing and nitpicking people who can do math.

The difference between your special ops analogy and the mathematics situation is that there is not a well defined special ops curriculum. Regardless of how much training you've inflicted on special ops troops, you can always choose to train more. With mathematics, there is a well-defined standard to complete (geometry covers topics A, B, C and algebra covers topics X, Y, Z ) and you're done. The curriculum needs to be exactly this hard and no harder (and no easier either). If the curriculum is at this appropriate level and students are "encouraged" to bail and go into law instead, then so be it. Watering down the curriculum to keep them doesn't do anybody any good -- not the students themselves or society at large. If this is truly the situation we're in, then we can train the few who are willing and able to complete this curriculum and continue to fill in the gaps with those trained in foreign institutions. That's not ideal, but it's hardly the end of the world. And it would give those few who are interested the tools to compete against the best that other countries have to offer on an equal footing. Can you imagine if we adjusted the difficulty of the medical curriculum to adjust to the supply of medical candidates? I'd be afraid to go to the doctor! Same thing that's true in medicine is true in mathematics -- there's a specified body of knowledge that needs to be covered, and our schools aren't covering it properly. That's the only point I've ever been trying to make.

PS, in my experience at a large, state university, most of the kids with Asian names are from America. Just look at the enrollment of Thomas Jefferson High School in Fairfax County Virginia (one of the three best public high school in the US). The school is 40% Asian. Also, I doubt that all of the Asian at UCLA or Cal-Berkley are fresh off the boat.

Well, that's not MY experience, or the experience of any number of fellow students who did graduate work in numerous types of programs (math, physics, astronomy, engineering) and types of institutions all across the country. Since you don't actually have any facts to contradict this fairly broad experience -- TJHSST may well be a counter-example but I couldn't actually find any numbers to confirm this (not that one example would constitute disproof of my general observation in any case) and your speculation regarding UCLA and Berkley isn't actually proof of anything -- I don't see how we can productively continue this particular line of discussion.

En Francais (In French)

I came across this education blog site whose content is very similar to mine. If there are any readers who read French (and I know there is at least one), you may want to take a look.

Filosofia Barata (Cheap Philosophy)

The capacity to learn is a gift;
The ability to learn is a skill;
The willingness to learn is a choice.

[original source unknown; I first saw this in Dune: House Harkonnen by Herbert & Anderson]

All of us have our God-given talents, so there's isn't anything I can do about the first sentence. As superdestroyer and allen have commented, there are strong cultural reasons why many students are unwilling to tackle math, science and engineering. While this is a serious problem, it is not one which I am particularly qualified to address. My interest lies mostly (and as far as this blog is concerned, entirely) in the second sentence: the ability to learn is a skill. Our high school (and increasingly, college) math programs need serious attention if they are provide those who are willing and able to tackle these fields the skills they need to be successful.

Response to Comment

Superdestroyer wrote...

How to you where the graduate students at the University of Chicago went to undergraduate let alone high school? Want to bet many of those Asian sounding names are Americans?

This is certainly possible. But my experience in graduate school would argue strongly against this. The students in graduate school with me with Asian sounding names almost universally did have foreign undergraduate preparation. And this phenomenom only becomes more pronounced when you climb up the academic food chain from the state schools I attended to the University of Chicago and other top schools.

My guess is that most of them just happen to come from social settings that emphasize hard work and do not look down at being "nerdy" but instead look down at being an "air head."

Well, then, that's an additional problem in our culture beyond what I've been discussing. But that is neither here nor there as to whether our high school and undergraduate math preparation is up to snuff. It's not. At the college level, our high school graduates can not compete with students trained in foreign high schools. To see evidence of this, all you have to do is look at the obscene number of remedial courses offered at even our "elite" colleges. And beyond that look at the courses offered which are not labelled remedial but which should be covered at the high school level.

• Introductory Algebra (wth is this anyway?)
• "College" Algebra (this is just a rehash of high school algebra)
• Trigonometry
• Pre-Calculus (I'm willing to cut a little bit of slack with this one, but really it belongs no later than the senior year of high school)

This then becomes a domino effect. At the graduate school level, our college graduates can no longer compete with students trained in foreign colleges.

My argument is that most of the upper middle class white kids who are capable of taking the math prep courses in undergraduate do not want to put in the hard work for majors in math or science because it would interfere with their social lives and their drinking.

I'm not disagreeing with you. I agree completely with this diagnosis. But I see this as a third problem (beyond my original point that U.S. math education is lacking and your earlier point that our culture looks down on "nerdy" behavior). I'm really most interested in the first problem. If as suggested by your comments (and I agree for the most part), cultural attitudes are such that math programs are not well populated by U.S. citizens, then so be it. But those programs at both the HS and undergrad levels need to have a higher level than they presently have!

As a final point, I will mention that you are very much mistaken if you think that if they wanted to, the upper middle class "white" kids (Latinos can be white too, you know) can still do well in a rigorous college math major after the sub-standard math curriculum they complete in high school. A few quick thoughts on this...

1. If you're not ready to take at the very least Calculus I (without preparatory "pre-calculus" courses and other such nonsense) your first semester in college, you will most likely never do particularly well in a math or physics major.
2. Without prior experience in difficult courses (in whatever field) requiring tons of homework, these students are often unable to handle the amount of work it takes to do even moderately well in college math courses. This is not generally a skill one develops on demand as soon as its need becomes suddenly apparent. Failure is the more common response to this situation.
3. Without the sharpening of the intellect that comes from an intellectually stimulating curriculum WAY before these students are even thinking about going to college, most of these students lack the necessary logical thinking abilities to be successful in such programs.

In my opinion, a sub-standard HS preparation in mathematics pretty much dooms these students to failure in undergraduate math, physics and engineering programs. Yes, the super-geniuses can always overcome this handicap and succeed. But the more average (who could have been successful in these programs if equipped with the right tools) will fail miserably.

Response to Comment

Superdestroyer wrote ...

It is not that the US students are not trained to compete, it is that they choose not to compete. Most suburban white kids would rather go into law, business, or government instead of grind it out against the asian kids in graduate school. Also given the long years of low pay to get a PhD in a science and the lack of job prospects, maybe the MBA or LLB are better choices, economically) than grinding it out in engineering school.

To an extent, this is correct. MBA and JD degrees are indeed financially better options; I've always viewed this as a premium that science PhD's pay for the privilege of doing what they love. However, there are two pieces of evidence that argue against fully accepting this theory.

1. Students who do choose to compete and try to get into graduate school in math, science and engineering often can't get in. It turns out that "the Asian kids" are better prepared and taking their slots. There can be no doubt this is a reflection of an inferior undergraduate preparation (unless one accepts the dubious proposition that Asian students are smarter, which I most definitely do NOT accept).

2. The lack of math preparation is a problem even among those who choose to get an MBA or JD.
(a) As an actuary, I've worked with several ERISA attorneys who are incapable of understanding basic mathematics (a serious impediment to someone who works with employee benefits plans). To them it was almost a badge of pride to admit, "I've always been horrible of math." Well, if they were better at it, they'd be better lawyers.
(b) In the business world, as you're probably aware, one of the largest new challenges facing U.S. corporations is derivatives and the appropriate management of risk. Yet U.S. students can not compete effectively for slots in Financial Mathematics programs. I urge you to look briefly at the list of current students in the University of Chicago's program.

"Hypotheticals"

As promised, I've been working on a paper addressing the NCTM Curriculum and Evaluation Standards for Geometry 9-12. However, there's a particularly appalling practice in which the NCTM engages which needs to be addressed separately. Chapter 7 of the PSSM (on Geometry) on p. 310 gives the following example of geometrical reasoning, which is presumed to follow from teaching geometry using NCTM principles.

The following proof demonstrates an ability to select and focus on important elements in the diagram, and it shows a solid understanding of the concepts involved and how they can be assembled to solve the problem.

First, I noticed that since AB and DE are parallel, angles B and E must be congruent. Also, angles ACB and DCE are congruent, since they are vertical. So now I know that the two triangles (ABC corresponds to DEC) are similar by angle-angle similarity. But that tells me that their corresponding sides are proportional. Since DE = 4(AB), I know that all the sides of triangle DEC are 4 times as large as the corresponding sides of triangle ABC, so CD = 4(15) = 60.

Now I just need to find the other side of triangle DEC to find its perimeter. But DF makes it into 2 right triangles, so I can use the Pythagorean theorem on each of those. FE^2 + 48^2 = 52^2, so FE is 20. (Actually, I just noticed that this is just 4 times a 5-12-13 triangle, but I saw that too late.) Then looking at CDF, this is 12 times a 3-4-5 triangle, so CF must be 36. (I checked using the Pythagorean theorem and got the same answer.) So the perimeter is 52 + 60 + 56 = 168.

Once I find the perimeter of ABC, I'm done. But that's easy, since the scale factor from DEC to ABC is 25%. I can just divide 168 by 4 and get 42. The reason that works is that each of the sides of ABC is 25% of its corresponding side in DEC, so the whole perimeter of ABC will be 25% of DEC. We already proved that in class anyway.

Wow! That’s the most impressive display of geometrical reasoning I’ve ever seen. I taught geometry for 6 years, and not even students who earned an A in my honors geometry/trigonometry class could have written this explanation. If this is an example of what NCTM math can accomplish, I’m sold. But wait a second… “Note particularly how the fictional student finds different connections to be sure her reasoning is sound.”

Incredible! They illustrate how their methods are supposed to work with a “fictional student”!? Has any REAL student ever written anything remotely like this? Can you imagine the outrage if doctors described how experimental cures would work on “fictional” patients?! And sadly this is hardly unique in NCTM “research.”

P. 311 – “The following hypothetical example illustrates how students might investigate relationships in a dynamic geometry environment and justify or refute conclusions.”

P. 342 – “Consider the following hypothetical classroom scenario”

I must admit I find it very difficult to take "research" like this seriously. How can I craft a logical response to a book that violates the most basic tenets of valid scholarly research so flagrantly? Maybe I should instead just concoct some "fictional" students who write dissertations in algebraic geometry after completing a traditional curriculum.

Response to Comment

Unknown Variable wrote...

Seems to me that this is a fine example of how we are a melting pot of nationalities and cultures.

To some extent, but don't you find it odd that all six members of the U.S. Olympic Chess Team are from one culture (the former Soviet Union)? It used to be said that U.S. players because Soviet players were true professionals who drew a salary from the USSR Sports Committee and did nothing but train for chess events, while U.S. players either (a) had other jobs so in a sense were amateurs despite their grandmaster standing or (b) had to go chasing tournaments to earn a living and thus did not have the time to train properly. Yet now the playing field has been levelled in this regard and U.S. players still can't compete. It definitely makes me wonder what's lacking in the background of the U.S. players.

Take a look at the nationalities of the players on the US National Soccer team. Would you say that because many of the players on the US squard are foreign born that there's an "athletic gap" between the US and the rest of the world? The same comparisons can be made for baseball. Would you say that we've become a country of "anti-athletes"? No.

Point taken.

The world is getting smaller. Other countries are catching up to the US' level for many things. Continuing with the sports theme. Take a look at basketball. I would say that the level of play for US players hasnt really changed for the last two decades, yet the NBA has more foreign born players than ever. Why? Because other countries are catching up to us in a game we've previously dominated. I think the same thing is going on here in the world of sciences.

Here I disagree, though. In my mind, the examples of chess, math and physics aren't really cases of others catching up to the US level, but rather cases where the US, already not the leader, is falling further behind.

Also keep in mind that over the last century (hell since the beginning of the USA), a great many of the scientific greatest accomplishments that are claimed as "US made" are from foreign born scientists.

True. What to make of this? Have US institutions always been sub-par? Certainly during the first quarter of the twentieth century, absolutely all the important math and physics was done in Europe. However, I have always believed that from around 1930 to around 1970 US preparation in these areas was on a par with European institutions. Maybe not?

...

By the way, unrelated to UV's comments, but I have been working on a paper discussing the NCTM standards in geometry. I hope to be able to post that here in the next week or so.

Anti-Intellectualism?

Continuing on the idea of my previous post ... This phenomenon does not appear to be limited to math, science and engineering. Take a look at my post on the "U.S." Chess Olympic team. Have we just become a country of anti-intellectuals?

More on the PhD gap

To expand on my earlier post regarding the "trickle-up" effect of bad math education on U.S. technical Ph.D. programs...

In the 1979–80 school year, for example, out of the total number of Ph.D. degrees conferred in the physical sciences, U.S. students received nearly 76% and foreign students received nearly 22% percent. (Percentages do not equal 100 because some students’ citizenship status is unknown.) In the 1996–97 school year, 57.5% of doctoral degrees in physical sciences were conferred on U.S. citizens versus 36.3% on foreign citizens. Furthermore, in that same year, of those receiving Ph.D.’s in mathematics and engineering, only 46.2% and 44.3%, respectively, were U.S. citizens whereas 46.6% and 49.5% were students with visas.

And this trend has continued since 1996-7. The bottom line is that students trained in the U.S. simply can not compete (of course, I mean in the aggregate when I say this) with foreign-trained students for the slots of our graduate schools.

Voyages

Thanks to Unknown Variable for bringing up the Voyages Curriculum. This curriculum provides an excellent example of NCTM math. Here's what their own website has to say about it. I seriously want someone to explain to me how a second grader can successfully master "algebraic thinking" without knowing the basic facts of arithmetic.

http://www.metrotlc.com/mtl_voyages.asp
Voyages • Grades 1-2

Metro recognizes what teachers have known all along: most students benefit from an instructional model that includes a variety of active learning strategies. Voyages’ two distinct lesson formats meet these diverse learning needs.

• Excursions lessons feature teacher-led, hands-on, real-life activities. These dynamic, interactive lessons typically take two to three days to complete.
• Anchors lessons develop critical mathematics skills and concepts while emphasizing the language of mathematics and algebraic thinking.

This is another website describing the Voyages curriculum. The following is the list of topics from the Grade 1 curriculum.

• Topic 1 DATA COLLECTION and ANALYSIS
• Topic 2 WHOLE NUMBERS and DECIMALS I
• Topic 3 GEOMETRY
• Topic 4 WHOLE NUMBERS and DECIMALS II
• Topic 5 MEASUREMENT
• Topic 6 FRACTIONS
• Topic 7 PROBABILITY

Some things to note quickly:
1) Data collection and analysis (including Organizing Data and Graphing Data) before such basic ideas as Numeration to 20, Addition Basic Facts, Strategies for Addition and Building Numbers from Tens and Ones? Are they insane?
2) Of course, the idiotic obsession with probability at all levels. Teaching likelihood to first graders?
3) Sub-topics like Addition Basic Facts, Strategies for Addition, Subtraction Basic Facts, and Strategies for Subtraction sound reasonable enough. However, as with many NCTM math concepts, such phrases often conceal much silliness. Is counting on your fingers a valid "strategy" for addition? My experience with NCTM curricula tells me the answer is yes, even though I have not worked with this particular one.

If anyone reading this has experience specifically with the Voyages math curriculum (or knows someone who does), I would love to hear from you. I am sincerely interested in seeing a set of these books, but I do not wish to support their program by buying them myself.

Response to Comment

UnknownVariable said...

"This book was written in 1955!" Yet we're still here. Still world leaders in technology, math, physics, etc. One might extrapolate that this book and your posts of similar vein are similar to chicken little exclaiming that the sky is falling... or one could imagine the accomplishments we could have made if real teaching reform took place back then (and now).

I think this comment warrants a response. First of all, I should clarify that I am in no way advocating a "sky is falling" scenario. Regardless of the final conclusion of this debate on NCTM math, life will go on more or less as it always had, for better or worse. I just feel very strongly that it will be better if NCTM math is wiped off the face of our school system and education colleges.

Second and more importantly, it is worth noting that we are still world leaders only to the extent that the U.S. is able to attract talent from elsewhere in the world. It has been a good many years since U.S. undergraduate education has been up to the caliber of undergraduate education elsewhere in the world. It used to be (not that long ago, 20-30 years maybe) that this statement could only be applied to high school education.

However, that sub-standard high school education eventually resulted in a sub-standard college education. It is now becoming increasingly apparent that this sub-standard college education is in turn starting to affect graduate education; it will not be long before my statement applies to graduate education as surely as it applies to high school education.

As just a quick example, look at Georgia State University's math and statistics tenured or tenure-track faculty (I chose GSU only because I knew where I could get my hands on this info quickly, but I am sure this is fairly representative of many U.S. institutions today).

Faculty members whose entire college education was at U.S. institutions...

• Jean Bevis - Ph.D. Mathematics, University of Florida, 1965
• Frank Hall - Ph.D. Mathematics, North Carolina State University, 1973
• Joseph Walker - Ph.D. Statistics, University of North Carolina, 1976
• George Davis - Ph.D. Mathematics, University of New Mexico, 1979
• Valerie A. Miller - Ph.D. Mathematics, University of South Carolina, 1985
• Ronald Patterson - Ph.D. Statistics, University of South Carolina, 1985

Faculty members whose college education was in part or in whole at foreign institutions (indicating highest degree earned abroad) ...

• Yu-Sheng Hsu - B.S. Mathematics, National Tsing Hua University, Taiwan, 1968
• Draga Vidakovic - B.S. Mathematics, Belgrade University, 1979
• Lifeng Ding - M.S. Applied Mathematics, Shanghai Jiao Tong University, China, 1982
• Guantao Chen - M.S. Mathematics, Huazhong Normal University, Wuhan, China, 1984
• Mihaly Bakonyi - M.S. Mathematics, University of Bucharest, Romania, 1985
• Zhongshan Li - M.S. Mathematics, Beijing Normal University, China, 1986
• Johannes Hattingh - Ph.D. Mathematics, University of Johannesburg, 1989
• Andrey Shilnikov - Ph.D. Mathematics, University of Nizhny Novgorod, Russia, 1990
• Susmita Datta - B.S. Physics, University of Calcutta, India, 1990?
• Yichuan Zhao - M.S. Applied Mathematics, Peking University, 1991
• Hongyu He - B.S. Mathematics, China, 1992?
• Imre Patyi - M.S. Mathematics, Eotvos University, Budapest, Hungary, 1995
• Alexandra Smirnova - M.S. Mathematics, Ural State University, 1995
• Florian Enescu - B.S. Mathematics, University of Bucharest, Romania, 1996
• Pulak Ghosh - M.S. Statistics, University of Calcutta, 1998
• Jiawei Liu - B.S. Applied Mathematics, Tsinghua University, 1998
• Gengsheng Qin - Ph.D. Statistics, Hong Kong University of Science and Technology, 1999
• Marina Arav - Ph.D. Applied Mathematics, Technion, Israel, 2000

In addition to the raw numbers, I urge you to notice the dates of the degrees awarded within each group. US graduate schools are increasingly becoming institutions where the best and brightest PhD graduates of foreign universities teach the best and brightest BS graduates of foreign universities. While certainly not a Chicken Little scenario, it certainly should worry us that the US will not be able to maintain its leadership position under these conditions. We need to be able to grow our own technical talent, and increasingly this is just not the case.

Retreat from Learning

I have just received a used copy of Retreat from Learning by Joan Dunn. From the flap of the book, "Retreat from Learning is a dramatic inside report on the failure of our public high schools ... shocking picture of lowered academic standards ... Joan Dunn's account of her teaching career is a story of four years of frustration and defeat in the face of the appalling waste of America's most precious resources - the minds and hearts of our youths."

This book was written in 1955!

Unsure how to proceed

After the initial elation at seeing my words on math education on the web and the subsequent flurry of activity of setting up this blog, posting material I've written elsewhere and trying to generate some buzz for this site, I now find myself unsure as to the best way to continue.

Analysis of the NCTM Standards requires long painstaking dissection and analysis. Writing up this analysis is not something that lends itself well to the blog format. I could concentrate on shorter observations that requre a less thorough comment on my part, but this is already done (and done well) by Joanne Jacobs. I see this blog as adding value to this discussion only if it does something different that is not done elsewhere. A couple of papers which illustrate exactly what I would like to do with this site are Some Thoughts On Constructivism and Some Disagreements With The Standards by Brian Rude.

These papers, at about 4000 words, are probably on the high range of the length that's possible in this medium. At this point, the only thing I can think to do is take some small aspect of the standards and write an analysis/criticism of it. (I've dusted off my copies of the Standards books and hope to be able to do this in the near future.) Then we can see if that style of writing could constitute a sustainable model for this blog.

Stay tuned!

NCTM Standards

NCTM non-members can gain 90-day free access to the full Principles & Standards for School Mathematics documents at http://standards.nctm.org/. I encourage everyone to do so and browse through these.

Two Examples

I wanted to zero in on two examples from the UPI article cited below that indicate exactly what is wrong with some of the instructional techniques of NCTM math. I liked these two examples because I think they illustrate what is wrong with NCTM math on both ends of the ability spectrum.

Cheney used an example from what she called an NCTM-inspired program called Mathland. Students were asked to solve this problem: "I just checked out a library book that is 1,344 pages long. The book is due in three weeks. How many pages will I need to read a day to finish the book in time?" The traditional method would be to divide the number of days (21) into the number of pages, getting 64. But, Cheney said, students today are often not taught long division. She held up a huge poster board covered with numbers, displaying the work of the student that Mathland featured as exemplary. "This particular student added up 21s until reaching 1,344," she said. Later in the program, NCTM President Lee Stiff said that he would never recommend such a method with numbers that large. But Stiff, a professor of mathematics education at North Carolina State University in Raleigh, said the technique is useful in teaching math concepts with much smaller numbers.

This is an excellent example of the problem at the low end of the spectrum. Yes, this will give you the right answer, but (a) understanding of what’s actually happen is nil and (b) if the assigned reading were 10541 pages and you had 83 days to get it done, the “technique” the student has learned is useless and he hasn’t learned an alternative technique that works in all cases. Prof. Stiff’s last comment in my mind completely misses the point – you want to teach the correct idea (division!) with small numbers, so that the student can confirm his understanding of what’s going on by doing something like this repetitive addition with small numbers. Then when presented the more complicated problem, he can just apply division (and UNDERSTAND why he's doing it). I don’t think this intellectual construct is beyond the ability of even below-average students, yet they are denied this true learning in lieu of mechanical meaningless manipulation. No wonder so many kids are growing up math-phobic.

[Edited to add: The mathematician in me couldn't let this go. It's worth pointing out that even conceptually adding 21 a bunch of times is meaningless. What's physically meaningful here is chunks of 64 (pages per day). Adding in increments of 21, while arriving at the correct answer, doesn't actually mean anything.]

In the 7th grade, he and his classmates were asked to find the area of a circle. Four weeks were devoted to the task. Traditionally, children were given the formula, but apparently these junior Archimedes were supposed to rediscover the uses of pi.

This is an excellent example of the problem at the high end of the spectrum. How many students among even our highest ability students are capable of deriving A = πr² on their own, even given all the time in the world? I took a course in topology and was able to understand and eventually replicate proofs of such famous theorems as the Urysohn Lemma, the Tychonoff Theorem and the Jordan Curve Theorem. According to NCTM math, I would be expected to construct this understanding on my own without a teacher lecturing to me. Not being a mathematician of the caliber of Urysohn, Tychonoff or Jordan, I would of course have failed miserably. Yet this is the same flawed idea that is used to justify asking 7th graders to derive A = πr². Absurd!

How the “experts” inflict NCTM math on us

NYC has inflicted an NCTM math program that goes by the name of “Everyday Math” into its K-12 curriculum. The research that justifies this program consists of papers such as the following two ...

On the other hand, Kamii and others demonstrated that students are capable of inventing their own effective and meaningful methods for computation (Kamii, 1985; Madell, 1985; Kamii & Joseph, 1988; Cobb & Merkel, 1989; Resnick, Lesgold, & Bill, 1990; Carpenter, Fennema, & Franke, 1992). Furthermore, these experiences were found to improve understanding of place value and enhance estimation and mental computation skills.

Well, that certainly sounds reasonable. Until we remember that “inventing their own effective and meaningful methods for computation” means adding 21 up sixty-four times.

It is also clear that the time saved by reducing attention to such computations [“complicated” paper-and-pencil computations] in school can be put to better use on such topics as problem solving, estimation, mental arithmetic, geometry, and data analysis (NCTM, 1989).

Again, that sounds reasonable for about two seconds. Until you ask yourself how someone who is unable to perform “complicated” (and the NCTM uses this term loosely) paper-and-pencil computations can perform mental arithmetic or data analysis.

Now, instead of reading all this eduspeak, I urge you to read one teacher’s experience with this curriculum.

No comment

http://www.foxnews.com/story/0,2933,134288,00.html

"Kids learn how to cook and they learn how to garden. They learn how to sit at the table and communicate with each other. To me, it is like an elementary education," she [Chez Panisse chef and owner Alice Waters] said. "It's more important that reading, writing and arithmetic."

[A nod to DVD for bringing this to my attention.]

Additional Evidence

One charge often levelled against me is that all this is purely anecdotal. Well, duh! I'm one person and I'm not a researcher in educational techniques; all I can do is discuss what happened to me personally.

For those who want to read additional material on this, I can recommend the following fascinating book: Ed School Follies: The Miseducation of America's Teachers by Rita Kramer. This book draws on the earlier The Miseducation of American Teachers by James Koerner (written in 1963!).

[edited to add author of the 1963 book - thanks to DVD]

What can we as individual citizens do?

Finnian said...

What is the solution though? Perhaps a better way to serve people would be to share your thoughts on how to resolve the issue.

What's the solution? I don't know exactly. If I had the answer, I'd be able to make billions, wouldn't I? I have some ideas...

1. Traditional methods work better than all of the "progressive" methods being experimented with. Certainly there are things that could be improved with traditional instructional methods, and technology has an important role to play (NOT giving calculators to 1st graders, though).

2. We need to ensure that teachers have a solid background in what they're teaching. I think elimination of all math (or whatever) education majors and replacing them with majors in math (or whatever) is a VERY good beginning. Teaching techniques are important to teach, but they should not replace content knowledge.

What can we do as individual citizens to fix it?

I attended at various times lectures by Jaime Escalante (the teacher on whom the movie "Stand and Deliver" is based), Ernest Boyer (who at the time was on the Board of the Carnegie Foundation for the Advancement of Teaching) and William Bennett (the former Secretary of Education). In each case, I posed exactly this question and did not get what I considered a satisfactory response from any of them. If they can't address this question satisfactorily, I'm not sure what I can say. But here goes...

1. Fight the school system. Attend school board meetings. When you hear the board members begin to accept this nonsense (and remember they are NOT educators, and generally don't know shit from shinola), speak up. I have seen public outcries stop this type of nonsense cold, most notably in 1992 in Miami, when they tried to slip constructivist nonsense into the elementary school curriculum as part of "Project Pheonix" - a rebuilding plan in the aftermath of Hurricane Andrew.

2. Be informed on educational issues. We are very much (too much, in my opinion) a country of credentials. We tend to bow to the opinion of "experts" even when those "experts" are the witch doctors (a nod to Richard Feynman) who bring us all these miserable failed educational experiments. Know enough about their absurd theories to be able to hold your ground in a debate.

3. Debate these ideas. Publicly. Loudly. Often. One of my friends (DAS - where are you? are you planning to pipe in here?) when he heard about this blog thought it was an excellent idea. He assures me that the best thing I can do is bring this discussion to light, and he has confidence that discourse, over time, is how we will eventually defeat bad ideas.

I’m not as confident myself that good ideas will always drive out bad ideas when both are open to the public scrutiny of open debate (if that were the case, would we be surrounded by so many bad ideas?), but I can certainly understand that the chance of this happening is much greater if there actually is debate than if the issue is ignored.

Someone like you, with insider experience, seems far more appropriate to do some good. Have you ever considered the bigger picture?

I consider the big picture all the time. But I’m not sure what I can do about it. I’m not an educational researcher, I’m not an expert on any of these educational/psychological subjects and I certainly don’t have the time to discuss this kind of thing (much less do something about it) on anything approaching a full-time basis.

On the other hand, I have given this a lot of thought, and I definitely have strong ideas on the best ways to teach most of this stuff. Someday I hope to have the time to develop my curriculum ideas into a full-blown set of textbooks, but that day is not yet here. My wife and I will probably home school when the time comes; that will probably provide the impetus to get this project started. For now, all I can do is the same as you – debate the ideas involved and hope that I can convince people that I’m right and all the NCTM math “experts” are wrong. I think that’s plenty “big picture” for now.

Let's agree on a new term

I finished going through the various links on the sites that discussed my original post, and I have found one worth reproducing here from the UPI (thanks to Jim Miller). Lynne Cheney suggests the use of the term "NCTM math." I like that. It has a couple of advantages:

• it clearly identifies the topic at hand - the NCTM Standards and those instructional techniques devised in connection with (although some might argue not always in agreement with) the Standards,
• it avoids this entire sidetracking argument that constructivism is a theory of learning as opposed to an instructional technique, and
• "NCTM math" can now be considered (at least this is how I look at it) to include constructivism as a theory of learning specifically applied to math (which in and of itself is not necessarily a bad thing), the specific instructional techniques used to implement the Standards, as well as the (sometimes dubious) overall educational philosophy nebulously connected to the Standards (such as giving first-graders calculators) which may or may not have any basis in constructivism.

What are we teaching the teachers?

Rudbeckia Hirta said...

I think your other anecdotes are much more telling and well-reflect what I see in my classes of pre-service elementary teachers: their content knowledge and ability for abstract thought is so weak that the pedagogical approach becomes irrelevant. In my non-teacher course I give review sheets like the ones you mention in another post -- without them everyone would fail, and I would get yelled at (failure rate is unofficially capped at 30% for my course). When they become teachers, they can't effectively teach constructively, as they do not possess the knowledge themselves. Before we spend too much time arguing about pedagogy, we really need to take a hard look at content and think about doing something about how little many teachers know (especially at the K-8 level).

I urge anybody who thinks that these comments are unduly harsh to look at the following books:
A Problem Solving Approach to Mathematics for Elementary School Teachers, Eighth Edition
Mathematics for Elementary School Teachers, Third Edition
Mathematical Reasoning for Elementary Teachers, Third Edition

Spend just a few minutes browsing through the pages of these books, and you will be genuinely shocked. Why are they teaching elementary school mathematics to people who are in college? Didn’t these people go to elementary school? Shouldn’t they have learned this stuff back then?

Seriously, if you have the opportunity to go to the bookstore of your local university, do so. Browse around; look at the textbooks in what passes for “higher” education in our teacher college classrooms. It’s scary!

Two comments

I wanted to address two related comments made to an earlier post I made.

Rudbeckia Hirta said...

The problems don't lie entirely with the NCTM Standards. Example, from the document itself (pages 5-6): "some of the pedagogical ideas from the NCTM Standards -- such as the emphases on discourse, worthwhile mathematical tasks, or learning through problem solving -- have been enacted without sufficient attention to students' understanding of mathematics content."

I agree to an extent. However, the problems with the standards is (a) they are generally so vague that they can (and do) mean anything that you want them to mean and (b) they generally end up meaning what math education professors want them to mean because nobody else has either the time or the authority to flesh out the details.

Chris C. said...

Constructivism is a theory of learning, and so it states that it's the way students learn (building on what they already know; contructing understanding) regardless of pedagogical specifics; even during a lecture or when the teacher is an authority figure.

I agree, but that brings two questions to mind.

First, if constructivism as a theory of learning (and it is), then we can conclude that as far as introducing it into the teaching of mathematics, it applies equally to lectures and other traditional techniques as to these new student-directed activities which are all the rage. In that case, we need to investigate what value it brings to the teaching and learning of mathematics. What are the insights that constructivism sheds on the question which will allow us to develop better teaching techniques? I have never seen this addressed in any research (well, U.S. research anyway, more on that in a minute). All I have seen is the quasi-religious belief that student-directed activities are intimately tied to constructivism in some way which the uninitiated can not hope to comprehend.

This ties directly in to my second point. Constructivism, as you point out, is not an instructional technique. However, in 6 years of teaching and 2 years of graduate education, I NEVER saw constructivism applied in any manner other than "Students construct their own understanding in geometry, so stick them in front of Geometer's Sketchpad." and "Students construct their own understanding in statistics, so stick them in front of a simulator." and "Students construct their own understanding in algebra, so give them blocks to play with to solve equations."

If anyone out there can point me to any applications of constructivism which are not of this nature PLEASE pass them along.

Discussion

I will be happy to discuss anything on this blog with those who comment. However, please note that implying I'm lying is not a good way to start a discussion. While no personal account can ever be 100% free from bias, I have made my best effort to present the truth in the stories below. In the interest of furthering this discussion, I will be happy to elaborate on, or clarify, anything for anybody who asks (nicely).

Constructivism

Constructivism: You can think of this as the new "New Math." It’s a theory of learning (which was making the rounds in the 90s when I was teacher and grad education student) that starts from the innocuous premise that knowledge is not taught by the teacher, but rather "constructed" in the mind of the learner. This statement is correct as far as it goes, but then constructivism devolves into absurd instructional techniques that remove the teacher as authority figure and turns him into a “first among equals” kind of role. Of course, this is absurd since the teacher knows the subject matter and the students don’t, but no matter. As you could expect, from this bizarre confluence of obvious theory and dubious application, you arrive at some of the most bizarre arguments I have ever seen, arriving at conclusions such as:
* first graders should be given calculators with which to learn arithmetic (news flash: they don't learn it - DUH!)
* students should do all work on a computer starting in middle school
* high schools should not teach algebra-geometry-precalc-calculus, instead focusing on statistics
I will probably elaborate later, complete with references to the National Council of Teachers of Mathematics curriculum standards, but the above is sufficient to follow the next couple of paragraphs.

I did some research on constructivist teaching of statistics. Basic statistics can be taught quite easily by explaining how to set up simple counting rules for the various possible results from an experiment, and then having students use these basic rules to solve increasingly complex problems. Statistics has been taught this way since it was invented. I learned it that way and the professor of math educ with whom I was working learned it that way, but the purpose of this research project was to inflict constructivism on some students and prove that they learned better than with traditional methods (yes, that's the way much education research is performed nowadays -- the conclusion is predetermined and then you look for evidence to support it). The student was given a computer and a (poorly-written) simulation program and asked to solve some problems by creating an appropriate simulation. (Oh, yeah, that will work! Students who do not understand basic statistics are *quite* adept at designing computer simulations.) Many problems could never be solved because the student stared at the computer, unable even to begin the simulation (and the teacher is not allowed to assist because that would put the teacher in an "expert" role -- excuse me, but that's what I want from my teachers!) Even in the few cases where the student set up a simulation, the answers meant nothing because the student did not learn any BASIC principle that could be applied in multiple situations. Each problem had to be approached from scratch. And of course, problems that could have been explained in 5 minutes (which I have done when I have taught stats) took 2 hours and the student still didn't really understand what had occurred. But the experiment was a success. Constructivism was somehow proved to be better, and the professor got a publication out of it.

My master's thesis in education. I designed the experiment. It involved four summer school geometry classes, two consisting of students who failed during the school year and two consisting of enrichment students who wanted to take it during summer so that they could take trigonometry or pre-calculus instead of geometry during the following school year. Two classes (one remedial, one enrichment) would be taught using traditional methods; the other two would be taught using constructivist methods. With my professor, I created the curriculum and learning objectives. I wrote the lesson plans for the traditional classes; my professor designed the constructivist exercises that would be used in the constructivist classes. I enlisted four teachers and two fellow students to assist me by doing the actual teaching (the constructivist classes required TWO teachers -- an indication right from the start that something was probably very wrong). For 6-7 weeks, I observed classes, I watched students performing the exercises, I interviewed students and teachers and parents, I evaluated assignments, I even eavesdropped on students once or twice. Each student in every class was evaluated twice, with a constructivist tool and with a traditional paper and pencil exam. The traditional students, from both the remedial and enrichment classes did very slightly better on the constructivist "final exam" than the constructivist students who had been exposed to this type of idiotic "critical thinking" exercise all summer, even though they had never been told anything about this type of assessment. And the constructivist students did ridiculously worse on the paper and pencil exam, with many scores very close to zero and almost nobody passing the exam, even among the enrichment group. For months after this exercise had reached a conclusion for me (as I'll relate in a moment) I felt absolutely horrible about what I had done to the constructivist groups. We gave a passing grade to every single student -- since everyone "passed" at least the constructivist exam even if not the paper and pencil exam. For the remedial groups, this probably was just fine since I'm sure they weren't really interested in learning anything, although I'm sure I intensified their math phobias (that's not just an opinion -- that conclusion follows from an analysis of my interviews with them). But what about the enrichment group? How many of them went into trig or pre-calc without the tools to be successful? Might some otherwise promising engineering or science students have been put off math forever? With the problem of grades solved, I began writing my thesis. The truth was that constructivism was proven to be a worthless pathetic failure. Of course, nobody ever gets a thesis accepted and published by bucking established theory, so I watered down my conclusion to a much weaker (but still true) statement: "No evidence was found that the constructivist methods are better than the traditional method." Professor read my first draft and turned it down outright. He told me to re-analzye my data so that I could state the constructivism was better or I would never graduate. So I walked out of his office and never graduated.

Education College Horror Stories

Scene: Introduction to Education class. Before midterm and final exams, professor handed out "review list" of 100 items. Exam had exactly these 100 items (written in exactly the same way, the order was even identical) as either T-F questions or fill-in-the-blank questions. Now that I think about it, there might have been some multiple guess questions, too. Another assignment in this class (worth 10% of our grade) was to keep a journal. Too bad blogs didn’t exist back then.

Scene: Undergraduate educational psychology class. First day of class after professor's lecture. We were assigned to work on a set of essays in class in groups (group work is really, really big in education schools). The first essay presented a classroom case study and asked something along the lines of "compare and contrast what theory A and theory B have to say about this situation and how each one might be applied to the situation" (I thought it was a pretty good question.) One student in the group proceeds to open up the book, copy out verbatim the textbook's explanation of theory A (in abstract, not applying it to the situation in the question at all) and then theory B. period. end of essay. Refused to listen to my objections that we had not answered the question. Thankfully, when I explained this to the professor, he allowed me turn in separate essays after that. Which turned out to be a good thing, since I received the only A in the class.

Scene: Honey, I Shrunk the Kids! Required watching (during class time, at that) in my undergraduate methods of teaching math course (required to get certified as a math teacher). Other assignments in that class: Watch Stand and Deliver (again, in class), writing instructions on how to get from one spot on campus to another (to see if you could write clear instructions, again the writing and following of these instructions was during class), writing a poem about how we felt about mathematics.

Scene: Undergraduate methods of teaching high school class. Professor (not a bad one at all, and positively brilliant by education school standards) gave us a list of 40 essays that would be related to (but not identical to) our final exam, which would consist of 10 essays. Students *demanded* that he narrow the list to only twenty essays and reproduce ten of them EXACTLY on the final; one of them threw a serious hissy-fit when the professor refused.

Scene: Social Studies Education doctoral-level class. Class was supposed to meet 2.5 hrs a day twice a week for eight weeks (total contact hours 40 hrs). Textbook was "Scandinavian Welfare States", but let's not even touch that. This story has other places to go. The first day, professor puts students into groups (of course) to work on a presentation about the book. Class will meet for 1 hr a week on alternate weeks so that each group can make its presentation. Total contact hours: 4! Sidebar -- Total contact hours involving actual work by the professor: 0!! Total work to be graded: one paper and one presentation turned in by each group. Exams: none. Additional reading: none. Keep in mind this is a DOCTORAL course. Final grade distribution: 12 A's!! (I wasn't actually in this class; I just had occasion to observe it while working next door in an education lab one summer.)

Scene: Math Education doctoral-level class: Grade based on two assignments: (1) Undergo a personal health improvement program and quantify the results (2) Predict the population of the US in 2030. Final grade distribution: 10 A's!! Side note: Nobody's report for the second assignment (except mine) actually used statistically valid arguments for extrapolating the population. And apparently nobody realized this as their answers varied from U.S. Census Department figures, sometimes by tens of millions.

Scene: Doctoral-level ed stat course that all Ed.D. and Ph.D. candidates in school were required to take. Some of the sections of this class were taught by and Adjunct Professor of Educational Research. The reason this person was an adjunct and not a regular full-time faculty member: did not have doctorate. Why not? Because after three attempts, had been unable to pass the College of Arts & Sciences statistical methods for social scientists course that was required to get your Ed.D. Keep in mind this person taught statistics!

Scene: Doctoral-level curriculum course. Entire grade based on final exam. Questions provided ahead of time. You could bring your pre-written answers into the exam room. Only requirement was that you had to copy them into the professor-provided blue-book in the exam time alloted. And of course, everybody got an A.

For the record, these events happened between 1990 and 1996 in three different universities.

Teacher Horror Stories

One day back in my undergraduate days, I was outside a Geometry class that had just finished waiting for the professor, who was my undergraduate advisor. Just leaving the room was a math-ed major, with whom I had taken a couple of classes, who wanted to be a high school math teacher. Actual live quote from said student while explaining that she did not like the geometry class: "When am I ever going to need this?" Well, I'm not sure, but maybe WHEN YOU HAVE TO TEACH IT?!

Picture a math exam room. You see several panicky students, attempting to memorize (rather than UNDERSTAND) basic middle school mathematics concepts like the transitive property. This is a scene we're all familiar with. The difference is that these were not middle school students; the room was filled with college seniors majoring in math education and math teachers. The cause of their concern: The dreaded mathematics subject matter expert exam. And yes, the word expert is used extremely lightly. The exam consisted of 100 questions, approximately in increasing order of difficulty. Elementary algebra made its first appearance around question 75, geometry around question 90. The last three questions consisted of basic calculus. Three hours were allocated; I took 50 minutes. When I left the room, I was able to notice that most of the candidates were still answering questions in the teens and twenties.

After I started teaching, the school where I worked had to dismiss two teachers because they could not become certified. The stumbling block: they couldn't obtain 840 on the SAT. Yes, that's 840 combined on the total exam, 40th percentile. One of them was an English teacher. Keep in mind that even if she got every question on the math section wrong (for a minimum score of 200), all she needed was 640 on the English section (80th percentile). So this ENGLISH teacher could not even perform on a standardized ENGLISH exam at a level that was higher than 80% of the students to whom she TAUGHT ENGLISH. I recently ran into a former student and in the course of conversation learned that this person (who could not break 840 on the SAT) taught the SAT prep course at the school. And sadly, just now while doing a quick Google search looking for some stats on the Florida Teacher Certification Exam, I learned that even this ridiculously low standard was dropped around 1999.

As far as graduate mathematics courses go, applied linear algebra is considered the easiest one in the curriculum. Math (as opposed to math-ed) graduate students tended to stay away from this one. It really didn’t go much deeper than undergraduate linear algebra – in fact at some schools our text (Linear Algebra by Strang) was used for introductory linear algebra. I took it because I was working at the time, and this class was offered in the evening. My friends who were math students basically made fun of me, implying I was looking for an easy A instead of trying to learn something. The class was filled with math education grad students. One of our assignments involved writing a MatLab program that took a bunch of points and determined the polynomial that went through all the provided points. A fellow student asked me to look at her solution before she turned it in. I spent 20 minutes attempting unsuccessfully to explain that her solution could not be correct because it didn't actually go through all the points. Eventually, she told me that something to the effect that she couldn't understand what I was saying and therefore I was probably wrong (how can you argue with that logic?) and turned in her homework as it was.

Reason for this blog

I was surprised to learn today that something I posted at http://www.actuary.ca/phpBB/viewtopic.php?t=17978 over a year ago seems to have surfaced in several places on the web in the last few days. I guess better late than never.

Some of the sites linking to and/or discussing my post are:
http://www.amritas.com/041002.htm#09282306
http://www.bundy223.net/~andyb/blog/archives/000215.html
http://lizditz.typepad.com/i_speak_of_dreams/2004/09/its_just_borken.html
http://genericconfusion.blogspot.com/2004/09/testing-education.html
http://www.seanet.com/~jimxc/Politics/ (search for constructivism)
http://www.joannejacobs.com/mtarchives/014443.html

The last one in particular has generated a fair amount of debate. I must say it felt really good to see my words generate so much discussion.

I would like to move this discussion forward, which has always been close to my heart even though I left teaching 8 years ago. The first thing I can do towards this end is to re-compile those stories and put them up here; a few of them probably require a little bit of editing before I do that, but I should be able to manage it in the next few days. Then maybe I can get some discussion going here.

New separate blog

I created a new blog for this whole education discussion. I didn't want this to clutter up my personal blog.