On NCTM Mathematics

Essays by William G. Quirk on various aspects of NCTM math:
Understanding the Original (1989) NCTM Standards
Understanding the Revised (2000) NCTM Standards
Understanding the Shallow Learning Expectations Promoted by NCTM Standards
Memorization and Practice are Ruled Out in NCTM Standards
The Anti-Content Mindset of the NCTM Standards
How the NCEE Limits K-12 Math

Truly excellent! My own views are almost exactly reflected by those articles. I have long wished to find the time to develop articles of similar depth explaining what is wrong with NCTM math.

FL bill would allow HS students to pick "majors"

The Florida House passed a bill that would require incoming high school freshmen to declare a major, just like college students. The bill faces an uncertain future in the Senate.

Students would be able to major in humanities, English, communications, math, science, history, social studies, arts, foreign languages and vocational skills. To graduate, students would have to earn 15 core credits in courses like math, science and English, 4 credits in major courses, and 5 credits in elective courses like drama and Spanish.

Source: CNN

I think this is a terrible idea! Our high school students graduate with little enough knowledge as it is. We are now going to remove 4 credits of general education and make it a "major"? What kind of "major" is a high school student equipped to handle? Backing up a step, how many high school freshmen are ready to choose a major at 14, when a considerable number of college students change majors?

Most College Students Lack Skills

More than 50% of students at four-year schools and more than 75% at two-year colleges lacked the skills to perform complex literacy tasks. That means they could not interpret a table about exercise and blood pressure, understand the arguments of newspaper editorials, compare credit card offers with different interest rates and annual fees or summarize results of a survey about parental involvement in school.

And that's not even the scary part of the story...

Overall, the average literacy of college students is significantly higher than that of adults across the nation!

Saxon Lovefest?

If you Google "Saxon Math Textbooks" to find information on this series of math textbooks, every link is breathless with worship of these books. I don't understand it.

I know some studies proved that the books were very effective with children from low-income families. Now, I am not a fan of the new new math or constructivism or putting lots of pictures in math books or any of that crap, but all the sites I read chastised state textbook adoption boards for not adopting these books. The fact is these books suck! Having used them in an actual classroom setting, I believe the claims that they work well with below-average students, but if you have an average (or God help you, an above-average) student, these books are worse than useless. I believe they are detrimental.

Ebonics

Incorporating Ebonics into a new school policy that targets black students, the lowest-achieving group in the San Bernardino City Unified School District, may provide students a more well-rounded curriculum, said a local sociologist. Mary Texeira, a sociology professor at Cal State San Bernardino, commended the San Bernardino Board of Education for approving the policy in June. Texeira suggested that including Ebonics in the program would be beneficial for students.

Note the subtle switch from "may" to "will be." Evidence? Dr. Texeira thinks so. A couple of her more liberal colleagues think so too.

Ebonics, a dialect of American English that is spoken by many blacks throughout the country, was recognized as a separate language in 1996 by the Oakland school board.

So a school board is qualified to decide what is or is not a separate language? Wow! Those guys are really good out there in Oakland. But maybe they should have deferred to the folks at the Linguistic Society of America, who disagree with them. An article which took a very balanced view of the Oakland decision can be found here. It (and other articles I have seen on this decision) ignore the question of whether Ebonics should be called or language or a dialect, since this is only a matter of semantics anyway. However, it disagrees with almost everything else in the 1996 resolution. Quite an interesting read.

Len Cooper, who is coordinating the pilot program at the two city schools, said San Bernardino district officials do not plan to incorporate Ebonics into the program. "Because Ebonics can have a negative stigma, we're not focusing on that,' Cooper said. "We are affirming and recognizing Ebonics through supplemental reading books (for students).'

Translation: Because Ebonics can have a negative stigma, district officials plan to incorporate Ebonics into the program WHILE CALLING IT SOMETHING ELSE.

"At every step we will see positive results,' [Board member Danny] Tillman said.

Only in education do we make such conclusions BEFORE the results are out. Hey, Danny, why don't we wait until the END of the year and then objectively analyze whether or not the results were positive?

[Texeira] said a child's self confidence is tied to his or her cultural identity.

Huh? It is? Why? Because she says so? What's the basis for such an absurd position?

She compared the low performance of black students to starvation. "How can you be angry when you feed a family of starving children?'

I disagreed with the previously quoted sentence, but I was at least able to discern some actual meaning in it. But here I am at a complete loss. What does this mean? Can anyone tell me what either the tortured metaphor or surreal question means? This is the kind of obtuse thinking that I've come to expect from the edu-left. Sentences like this do not betray an inability to communicate as is often suggested; they betray an inability to think.

[The complete article can be found here. Thanks to DVD for bringing it to my attention.]

Question

http://www.onlinemathcourses.org

Middle Mathematics
Algebra I
Algebra II
Geometry
Pre-Calculus
Calculus I
Calculus II
Probability & Statistics

Each course awards 3 semester hours of graduate credit (500 level or higher) from the School of Graduate Studies.

Can someone explain to me why a course in high school algebra merits 3 graduate credits?

Computer Education Research

A topic interesting to me in a personal way, since Cobb County Schools where I live are talking about spending $70 million to get every student a computer.

http://www.csmonitor.com/2004/1206/p11s01-legn.html

From a sample of 175,000 15-year-old students in 31 countries, researchers at the University of Munich announced that performance in math and reading had suffered significantly among students who have more than one computer at home. And while students seemed to benefit from limited use of computers at school, those who used them several times per week at school saw their academic performance decline significantly as well.

Computer technology "is used too much and very unwisely in the younger years, and not wisely enough in the older years," says Todd Oppenheimer.

Georgia Education System Challenged in Court

Fed up with his options, Williams joined two other Atlanta parents in filing a lawsuit Thursday in Fulton County Superior Court, asking that the state's education system be declared unconstitutional. The parents seek fundamental changes that would greatly increase their educational choices. The suit suggests several ways to accomplish this, including new attendance rules allowing parents to send their children to any public school, no matter where they live. It also proposes that parents be allowed to use tax money to help pay private school tuition, though the hot-button word "vouchers" is never mentioned. The parents also suggest the virtual elimination of regulations that have limited the number of charter schools.

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.