Teaching Math
So I've been thinking a little more about education some, after some off-topic discussion on an RPG forum, "Fuzzy Math?".

My limited teaching experience is in some grading on the undergrad college level (math and physics), some physics laboratory instruction and help room supervision, a bit of substitute teaching of 5th-6th grade computer science, and a tiny bit of lecturing. I was a member of the American Association of Physics Teachers, and went to their 118th national meeting in Anaheim.



Background

More broadly, I've often been curious has been about how our education system has been progressing. The U.S. National Assessment of Education Progress has collected statistics by giving identical tests over time since its founding in 1969. This is not the same as the the regular tests like the SAT that regularly change, but rather a sampling measure that strives to remain constant. If you want quick info, you can skip to the online executive summary from 1999. The full article has more discussion of methodology. For the even shorter summary:

Generally, the trends in mathematics and science are characterized by declines in the 1970s, followed by increases during the 1980s and early 1990s, and mostly stable performance since then. Some gains are also evident in reading, but they are modest. Overall improvement across the assessment years is most evident in mathematics. National trends in average reading, mathematics, and science scores are depicted in Figure 1.

Two controversial movements in teaching math have been around over the past few decades. In the 1960s, fears about the dominance of Russian mathematicians and physicists motivated the New Math trend -- where a number of advanced math concepts were introduced early, like teaching commutation as a principle to 2nd-3rd graders. As a physicist, I find some of New Math quite fascinating since I can see how parts of it were directly taken from field theory.

The more current controversy is over "reformed math" or "constructivist math", which several sources date back to a report by the National Council of Teachers of Mathematics in 1989. There is a 17 page short version of their recommendations posted on the Department of Education website here: "Curriculum and Evaluation Standards for School Mathematics". There was a longer report from a few months later, still available paperback on Amazon, that runs 258 pages.

From a roughly look at the NAEP data, it appears that trends concurrent with New Math were unsuccessful -- at least, there was a moderate downturn from 1973 to 1982 -- though really that may have been after the heyday of New Math. The
reformed math effort is only moderately successful. There have been gains in math scores since the eighties. However, most of the gains are between 1986 and 1990, and I think the NCTM recommendations in 1989 are unlikely to have had their full effect by 1990.

Here in California

The current California content standard for mathematics was adopted in 1997. The current California mathematics curriculum framework, adopted in 2005, is also available. Appendix D has sample problems for all grade levels. The latest test scores statewide are also available.

Basically, the current California curriculum is fairly traditional. I believe that this was a shift of several years ago. It considers some of the NCTM recommendations, but does not strongly pursue them -- compared to some of the more controversial curricula.

Controversy

In the thread, John Morrow cited news articles such as the 2001 Time Magazine article "This is Math?" by Romesh Ratnesar; and the April 2000 New York Times article, "The New, Flexible Math Meets Parental Rebellion" by Anemona Hartocollis. These were extremely critical of the new curricula based on the NCTM recommendations, but did not cite any widespread drop in scores. The Time suggested mixed results -- In a few states that have emphasized new-new math, such as Connecticut, there are early indicators of improved student performance. Critics in California, on the other hand, point to test scores in cities like Santa Barbara and Palo Alto that show at least temporary drop-offs after whole math has been introduced.

The articles also cite some examples of approaches clearly pitched as "horror stories", yet most of these seemed perfectly reasonable to me. In my experience, interactive and lab sessions are often more rewarding in learning than lectures. At the college level, it seemed to me that they were used less because they demanded more individual attention. Lab sections had only a dozen students and were lead by graduate students, while full professors would lecture to 50 or more students.

What the NCTM Really Says

Here are some key quotes from the NCTM's 1988 "Curriculum and Evaluation Standards for School Mathematics"...

For many students the study of mathematics begins and ends with computational skill; mastery of pencil-and-paper procedures is believed by many to be prerequisite to the investigation of applied problems, algebra, geometry, or other mathematics. Those students not demonstrating an early ability at fast and accurate computation are often relegated to remedial classes dominated by repetitive drill. At the secondary level these students typically end their mathematical studies by the ninth or tenth grade, denied the opportunity for many careers.

In order to correct imbalances in the mathematical content studied by students, the report proposes that schools: 1) require all students take 12 years of mathematics (K-11; 13 years is suggested, particularly for college-intending students); 2) implement a core curriculum that allows all students the opportunity to study the important ideas and methods of mathematics; and 3) remove pencil-and-paper computational ability as a necessary prerequisite to the study of other mathematics.

The standards report states bluntly that no basis exists for the belief that pencil-and-paper skills are prerequisite to the study of other mathematical ideas, recommending that all students be exposed to essentially the same topics -- a much broader set than is currently included in any of the present tracks. Obviously, not all students need to, or will be able to, study these topics at the same level of mathematical rigor. And, in adding a wide variety of new topics to an already crowded curriculum, something must go.

The proposed solution to differing ability in the classroom is to vary not the content, itself, but the "depth and breadth" in which content is covered. For examples, all tenth graders would study some geometry -- some would study geometry at an informal level emphasizing spatial visualization and simple problem situations, others would experience a more formal, axiomatic approach including a stronger emphasis on formal proof and introduction to non-Euclidean geometries.

Some things stood out to me from it. The report is clearly in reaction to some prior practices where topics like geometry or word problems relating numbers to real-world situations were considered "advanced". It implies that students wouldn't get to them at all until after years of only drilling through doing addition, subtraction, multiplication, and division on paper -- sometimes not ever getting to anything else.

My Two Cents

The basic recommendations of the NCTM seem reasonable to me, and they reference studies on the effectiveness of teaching methods. I'm not sure about all of them, but the test results seem to be fair although they do not show marked improvement.

One issue it emphasizes is how different the world of today is. In the 1960s, the skills of working out multi-digit computations on pencil and paper were absolutely vital to the workforce. Those were core skills used by people at their jobs every day for critical tasks, and a slide rule was an important instrument. I remember being amazed when watching the movie Apollo 13 to see astronauts sharpening their pencils to calculate their changed orbit. It was good sense for this to be emphasized in schools at the time, and it prepared students well for the world they would enter.

However, the necessary skills of working adults today are notably different. Adults are rarely, if ever, called on to do multi-digit operations with pencil and paper. Conversely, there are a lot of key mathematical skills used every day by people now that were rare or non-existent in 1970. Someone who goes to work in an office will not be asked to add up columns in a book, but rather to do tasks like sorting accounts in a spreadsheet and making a graph of projected growth.

The objections seem to be that fast/efficient pencil-and-paper operations are a critical prerequisite skill for any other mathematical understanding -- a contradiction to the NCTM's assertion. Personally, I don't buy it. As a physicist, there was an old joke:

A reporter asks a mathematician, a physicist, and an accountant "What's 3 times 3?" The mathematician thinks for a moment, and replies "An answer exists... and it is rational." The physicist shrugs and says "It's order 10." The accountant smiles and asks, "Well... what would you like it to be?"

It's a joke I've told a lot because it felt very true. Physicists are used to having 10% or more margins of error, and thus skipping exact calculation in favor of estimation.

I'm not completely set on the specific changes recommended by the NCTM, but they aren't unreasonable. They don't seem to have improved things majorly, but by the numbers I've seen they seem moderately successful.