In the Feb/Mar 2009 MAA FOCUS, current President David Bressoud's column is entitled, "Mind the Gap." In response, Dom Rosa, former associate professor of mathematics at Teikyo Post University in Waterbury, Connecticut and full-time crank, posted to email@example.com:
According to the article: "For four-year undergraduate programs, calculus and advanced mathematics enrollments dropped from 10.5% of all students in 1985 to 6.36% in 2005. This occurred while high school students were taking ever more mathematics at ever higher levels."
This should not surprise anyone who is minimally aware of the current state of pseudo-education in the U.S. I keep meeting more and more students who do nothing more than scribble on photocopied handouts that are distributed in so-called Geometry, Algebra II, and Precalculus courses. They never open their bloated doorstops. Recently I met a student who was told, "You don't need your [junk] book; just leave it at home."
The article also points out: "... more students arrive at college having earned credit for Calculus I, but they have not produced larger enrollments for Calculus II. Over these same 20 years, Fall term enrollments in Calculus II dropped from 115,000 to 104,000. Across the board, students are arriving at college and failing to take the next course in their mathematical progression."
My question is this: How is it possible that Bressoud who is the current MAA president, his predecessors, and other hierarchs appear to be so clueless about the mathematical pseudo-education of American students? It seems to me that this is where the real "Gap" exists.
This post drew a couple of rather skeptical replies, the first from Metropolitan State College of Denver mathematician Lou Talman, who asked:
Dom, have you considered the possibility that "the current MAA
president, his predecessors, and other hierarchs appear to be so
clueless about the mathematical pseudo-education of American
students" because *you* perceive a problem that isn't there?
And Lou's response engendered the following from anti-progressive Internet ghost, Haim Pipik (aka, Edmund David):
The premise of the argument is,
"According to the article: 'For four-year undergraduate
programs, calculus and advanced mathematics enrollments
dropped from 10.5% of all students in 1985 to 6.36% in
2005. This occurred while high school students were
taking ever more mathematics at ever higher levels.'"
Do you accept the above as an accurate representation of the facts? If this is a fact, do you accept that declining enrollments in the calculus is a problem?
Dom thinks this declining enrollment is a problem, and he asserts that the source of the problem is bad teaching in K-12. If you agree it is a problem, what do you think are some of the major contributing causes?
It's simple enough to dismiss Dom Rosa as a conspiracy theorist: he has a track record of posting and writing letters to various media outlets about his three favorite themes: the pseudo-education of America's youth in mathematics; "doorstop" textbooks; and various "rackets" in education, from standardized testing to calculators to any notion, book, method, or tool conceived after his beloved golden era of learning math in Massachusetts, a time when, if he is to be believed, teachers, books, and students were all simply wonderful, and everyone graduated with a genuinely deep, rich knowledge of mathematics. The question of how we still managed to produce yet another generation of Americans who fear and loathe mathematics during that period never seems to cross Dom's mind, or, if it does, he ignores it.
Nonetheless, the far less "cranky," but far more dangerous neo-conservative cum Libertarian, Mr. Pipik, uses Dom's latest bit of crankery to once again raise the specter of evil on the part of America's public schools. For those not playing along at home, Pipik's recurrent themes are that: we spend too much money on public education, much of it wasted, mis-spent, pilfered, given inordinately to lazy teachers and administrators who fail to show accountability and don't produce high test scores, etc.; the private sector would do a much better job of educating kids; we're wasting time and money trying to teach mathematics to most kids, who really can't learn it, don't want it, and don't need it; and anyone who disagrees with him is stupid and a communist who loves Stalin, Mao, and wants to eat the rich (well, maybe I'm exaggerating with that very last part).
Thus, while I don't think Dom or Haim has much of a case, I offer the following serious reply:
My ReponseHaving just read Bressoud's piece, I'm wondering if either Dom or Haim bothered to do so. Had they actually read it, they'd see that Bressoud gives us a good deal to chew over, all of which needs to be looked at carefully, not skimmed to find the one or two tidbits that can be spun just the way the reader has already concluded the "truth" requires.
It's necessary to look at Bressoud's analysis paragraph by paragraph to get a realistic picture of what real questions it raises:
DB: “Mind the Gap” is an appropriate metaphor for one of the greatest challenges facing undergraduate mathematics education today. There is a significant gap between students’ experience of mathematics in high school and the expectations they face on entering college, and there are troubling signs that this gap may be widening. There are serious problems in K–12 mathematics education, but college faculty also need to look to their own house and think about the first-year experience of their own students.
MPG: It's a safe bet that Mr. Rosa is NOT looking to his own house. He never does. And Mr. Pipik? We don't even know what his house might be, but he rarely criticizes departments of mathematics (too many Mathematically Correct and HOLD allies there to risk doing so). No, his focus, like Dom's is always on the other guys: K-12 public school teachers, not because he gives a rat's patootie about how much math gets taught or learned by most kids, but because of his political agenda: destroying public education and turning things over to the private sector. But since he has nothing to contribute to any conversation about how to actually improve teaching at any level in any subject (unless being snide and smug count as tips for bettering things), he can be safely ignored for the duration of this analysis.
DB: In my article “Is the Sky Still Falling?” (2009), I observed that four-year college mathematics enrollments at the level of calculus and above declined from 1985
to 1995 and have since recovered to slightly below the 1990 numbers.
MPG: Well, let's be sure to gloss over that last statement. I hate to be the first person to point this out, but there are cycles in most areas of study. And the reasons vary enormously from cycle to cycle. But if there was a drop from 1985 to 1995 and then a recovery to just below 1990 levels, is the sky actually falling or merely moving up and down rather lazily? And one might assume that four year colleges, not community colleges, represent the majority of stronger, better prepared students.
DB: Two-year colleges saw calculus enrollments rise in the early ‘90s, then fall to well below the 1990 number, while the number of their students requiring remedial mathematics exactly doubled. In percentages, the picture is dismal. For four-year undergraduate programs, calculus and advanced mathematics enrollments dropped from 10.05% of all students in 1985 to 6.36% in 2005.
This has been debated and discussed widely. Bressoud doesn't offer any real analysis here. No hypothesis, just numbers and the word "dismal," which of course implies a great deal but tells us nothing. If enrollments in community colleges are growing rapidly, it's quite conceivable that what this means regarding the PERCENTAGES of students who enroll in calculus and beyond SHOULD be dropping. Indeed, it would almost have to drop, because we're getting vastly more students in such colleges who are coming from the lower echelons of high school graduates. It does not follow from his statements that things are any worse in terms of the qualifications of comparable percentile ranks of high school graduates today and at various points in the past. Until someone actually comes up with that sort of comparison, with concrete examples to illustrate the nature of any apparent decline or growth, we really don't know squat based on the above other than that more kids are enrolling in two year colleges but a lower percentage of them go into higher math. And when one considers who goes to these schools and why, this seems utterly NON-dismal.
DB: This happened while high school students were taking ever more mathematics at ever higher levels. In 1982, only 44.5% of high school graduates had completed mathematics at the level of Algebra II or higher. By 2004, this had risen to 76.7%. In 1982, 10.7% had completed precalculus. By 2004, it was 33.0%, over a million high school graduates arriving in college ready — at least in theory — to begin or continue the study of calculus. Yet over the years 1985–2005, Fall term enrollments in Calculus I dropped from 264,000 to 252,000.
MPG: Hmm. Why do I not find myself panicking over a drop in freshman calculus enrollment of 12,000 students? Aside from many issues that even Bressoud touches on below, anyone paying attention may have noticed that we're not exactly dying for professional mathematicians in this country. Any opening to teach mathematics at the post-secondary level has dozens of qualified applicants. In some parts of the country, it's no easy task to find a public school that is hiring in grade 6 - 12 mathematics classrooms. Engineers are getting laid off in many disciplines. No general cry for more physicists has gone out. Straight A's in four semesters of calculus is no guarantee of admission to medical school, even less so into veterinary school (though exactly why those courses are prerequisites for either program remains a bit of a mystery). That 12,000 student drop, which remains unanalyzed as to possible causes, doesn't seem linked to the current economic crisis: indeed, one might reasonably assume that all those brilliant economists and MBAs took lots and lots of calculus.
DB: Admittedly, many more students today arrive at college already having earned credit for Calculus I, but they have not produced larger enrollments for Calculus II. Over these same 20 years, Fall term enrollments in Calculus II dropped from 115,000 to 104,000. Across the board, students are arriving in college and
failing to take what should be a next course in their mathematical progression.
MPG: It would be important to know if Bressoud's figures account for two things: first, how many of those students who come to college with a semester's credit already in hand wisely choose to wait until second semester to take Calculus II with friends and to give themselves an easier first semseter in college, something I have frequently recommended to high school seniors. There isn't a ticking clock, and there are lots of other courses in which a bright student might wish to enroll regardless of whether s/he is ready for second semester calculus in the fall of freshman year; and second, what percentage of those students NEVER take another mathematics class at the level of calculus or above? Without providing information on either of these concerns, Dr. Bressoud really has no grounds for concern on the above numbers, yet he seems to imply we should be terribly worried.
DB: The college community is not blameless. Too many good students are turned off by their initial college experience in mathematics. Too often, first-year courses are large and impersonal, instructors — especially adjunct faculty and graduate teaching assistants — are under-prepared, and little thought has gone into implementing appropriate pedagogies. Moreover, a common complaint that I hear from high school teachers is that colleges focus exclusively on what students do not know, with the result that many students find themselves assigned to classes they find stultifying.
MPG: I'm not willing to put too much blame on adjuncts or graduate students. I have had some very good instruction from some of them. At the University of Michigan, one of the finest mathematics professors I studied with was a post-doctoral student. When I decided to take a pair of refresher summer classes in calculus, both instructors were graduate students and were very clear, concerned and competent instructors. On the other hand, one of the most horrid mathematics classes I had was taught by a full professor. He lacked just about every requisite skill imaginable for good teaching other than subject matter competence.
That said, Bressoud would have it right and do students a huge service if he didn't try to foist the responsibility for bad pedagogy on lower-rank teachers and admitted that there is a vast amount of bad instruction in mathematics departments regardless of the rank of the teacher. Of course, there are also outstanding mathematics professors of higher rank (and I don't merely mean great mathematicians who can communicate with three or four truly gifted students, but the Polyas, the Edward Burgers, and others who are deeply committed to teaching mathematics to as many students as they can possibly reach.
DB: This last is a tricky issue. The answer cannot be that colleges lower their expectations of what it means to know algebra or calculus. It does mean that colleges need to rethink how to get students from where they are as they enter college to where they need to be. It does mean offering more routes into good mathematics and restructuring existing courses so that they acknowledge and build upon what students do know while remaining mindful of and addressing the gaps in this knowledge. Especially when a student needs to relearn a topic that appears familiar, we must ensure that the course is structured so that it provides fresh challenges that entice students to keep moving forward.
MPG: No argument with the above, especially if Dr. Bressoud can make a commitment for the professional mathematics community in post-secondary institutions to communicate the above to parents, employers, K-12 educators, administrators, guidance counselors, college admissions officers, politicians, and the media. Further, to try to minimize the counterproductive activities of the small, vocal minority of activists within or affiliated with the professional mathematical community who are set on preserving the status quo at all costs and who actively oppose any sort of reforms along the lines he mentions. It is high time that both major organizations of professional mathematicians took a principled stand against groups such as Mathematically Correct and NYC-HOLD.
Additionally, if Dr. Bressoud is serious about creating alternative routes and attractive options for students without sacrificing mathematical standards, it is high time that the professional mathematics community support teaching discrete mathematics, the underlying mathematics of computer science, and powerful connections to technology and the Internet as important paths for youngsters to explore and study before they reach college. The notion that calculus is the exclusive meaningful goal for K-12 mathematics education is an outmoded idea that will be modified in the minds of the public and teaching community only when it comes with the support of professional mathematicians.
DB: We have learned a lot about teaching undergraduates in the past 20 years. There are proven programs for bridging the gap. The Emerging Scholars Program is one. Stretching Calculus I over two terms with precalculus topics treated on a just-in-time basis is another. But there are no magic bullets. Each college and university must examine what others have done and adapt to its own situation those programs that are most appropriate.
MPG: I can agree with the statement about the lack of panaceas. But if the focus remains solely on getting students into calculus, all the other fine things Dr. Bressoud mentions will simply break against the walls of tradition, in all likelihood. As long as mathematics is viewed so narrowly through the lens of analysis, the traditionalists will feel they still have a mandate to demand control over the K-12 curriculum. The preceding sixteen years of the Math Wars show clearly that the hard-core educational conservatives will never concede an inch towards meaningful reform of mathematics teaching at ANY level, K-12 or post-secondary, as long as they can promulgate the idea that they represent the entirety of professional mathematicians or nearly so.
It is incumbent upon anyone in Dr. Bressoud's position, if s/he wishes to make positive changes feasible, to be much more careful in the examination, analysis and interpretation of statistics such as those cited in "Mind the Gap." While few people doubt or deny that we can continue to improve the quality and effectiveness of mathematics education at all levels in this country, the sky really is not falling. American kids are neither more stupid nor more ignorant today than in the past. Nor are they being "pseudo-educated." They could use, however, better guidance from the professional mathematics and mathematics education communities about what mathematics really is and the many ways in which they could positively engage with it. As always, the question is whether those who understand the need for change have the courage to stand up for it, even against respected colleagues desperate to stand in its way.