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?