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发帖数: 1 | 1 The rush to calculus is bad for students and their futures in STEM
November 20, 2015 6.12am EST
Author
Kevin Knudson
Professor of Mathematics, University of Florida
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The author, teaching at the very front of his calculus class. Kevin Knudson,
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Two years ago I taught a section of Calculus I to approximately 650
undergrad students in a large auditorium. Perhaps “taught” isn’t the
right word. “Performed,” maybe? Unsurprisingly, my student evaluation
scores were not as high as they usually are in my more typical classes of 35
students, but I do remember one comment in particular: “This class
destroyed my confidence.” According to a new report from the Mathematical
Association of America (MAA), this outcome is common, even among students
who successfully completed a calculus course in high school. So what is
going on?
Former MAA president David Bressoud led this five-year comprehensive study
funded by the National Science Foundation. He’s been thinking about this
problem for many years and has synthesized a huge amount of data measuring
high school and college calculus enrollments. I heard Bressoud speak about
some preliminary results of the study a few years ago, and one piece of data
stuck in my head: in the mid-1980s, when I was in high school,
approximately 5% of high school students took an AP exam in calculus.
That aligns with my personal experience in which there were about 150
students in my entire North Carolina county taking calculus in any given
year (out of roughly 3,000 high school seniors). Nationally, about 60,000
students took an AP calculus exam my senior year (1987). Today? That number
has risen to nearly 350,000 students taking an AP exam in calculus in 2011 (
roughly 15% of high school students). As one of my colleagues remarked after
Bressoud’s talk, it’s not as if the talent pool has gotten that much
deeper in the last 30 years. This tripling of the proportion of students
taking these exams feels wrong somehow.
Why the dramatic increase?
There appear to be at least two driving forces behind the rush to calculus.
Transparent, research-based, written by experts – and always free.
Breakdown of all Advanced Placement exams taken in 2013. Ali Zifan, CC BY-SA
One is college admissions. Students and their parents seek an advantage in
the increasingly competitive admissions tournament, and the number of AP
courses taken is a metric that is easy for students to boost. The increase
in the number of AP exams taken is not unique to calculus; indeed, the total
population of students taking exams doubled between 2003 and 2013, with the
number of exams administered increasing by 150% over that period. As the
name “Advanced Placement” suggests, these exams often yield college credit
for students; this appeals to parents, as well, since it ostensibly lowers
tuition costs later.
Another factor that must be considered is the overall decline in support for
enhanced education for gifted students. In an era of shrinking education
budgets, school administrators find it tempting to conflate advancement with
enrichment. Pushing gifted students ahead at a faster rate via AP courses
is seen as a solution for meeting the needs of advanced students.
This approach may be dangerous in any discipline, but it is especially
problematic in mathematics, where a strong foundation is key to success in
upper division courses. The general strategy in high school is one of
uniform advancement – taking advanced coursework in all disciplines under
the assumption that gifted students are exceptional in every subject. In the
drive to make it to calculus by the senior year, students often rush
through algebra and geometry in lockstep with their gifted peers whether
they are ready for it or not.
The end result is a group of students who have “succeeded” in high school
calculus without really having the proper foundations, a tower built on sand
. It is quite possible for students to learn the mechanics of many
categories of calculus problems and to answer questions correctly on exams
without really understanding the concepts. To quote the MAA’s report:
In some sense, the worst preparation a student heading toward a career in
science or engineering could receive is one that rushes toward accumulation
of problem-solving abilities in calculus while short-changing the broader
preparation needed for success beyond calculus.
Calc students’ favorite friend: the graphing calculator. Gene Wilburn, CC
BY-NC-ND
College versus high school calculus
There are two flavors of AP calculus, AB and BC. The former is equivalent to
a typical first-semester college course, while the latter covers the first
two semesters. Exams are scored from 1 to 5; most universities grant credit
for a score of 3 and up.
Many students take Calculus I again at their universities, even if they have
a passing score on the AP exam. There are many reasons for this: some
colleges insist (engineering programs in particular) and many medical
schools demand that applicants take the course at a university. Or students
may not feel particularly confident about their abilities. In my own
experience, the number of students retaking the calculus course is very high
– in a typical section of engineering calculus, up to 90% of my students
have taken it in high school. While there are some positive aspects to
retaking the course, there are downsides, the most notable of which is
overconfidence and a student’s misplaced certainty that he or she already
knows the material.
A typical first-semester calculus course consists of 45 lectures delivered
three times per week over a 15-week term. The pace is quick. Contrast that
with a typical high school Calculus AB course, which meets five days per
week for 180 class meetings. The college course covers the same material in
a quarter of the time; students must therefore have solid skills in algebra
and geometry along with good study and work habits to succeed.
So this is the crux of the problem: students lacking the requisite
foundational abilities may not succeed because the college faculty member
expects them to be at ease with these more basic ideas, freeing them to
absorb and understand the new, more conceptual material. The rush to AP
Calculus has instructed students in the techniques for solving large classes
of standard calculus problems rather than prepare them for success in
higher mathematics.
It’s precisely this disconnect that causes students to lose their
confidence if they don’t do well in university calculus. All through high
school, the evidence suggested that they were “good at math” because they
succeeded in parroting what they saw demonstrated in class. Parroting is not
learning, however, and may hide a student’s true abilities.
What to do?
The authors of the MAA report sum it up best:
Students are better prepared for post-secondary mathematics when they have
developed an understanding of the undergirding principles which, when
accompanied by fluent and flexible application of the concepts and
procedures of precalculus mathematics, enable them to understand calculus as
a coherent and broadly applicable body of knowledge.
Like so many issues in K-12 education, the reasons that we have gotten to
the current state are manifold, and reversing trends is difficult. But if we
want to advance STEM education and continue to produce a high-quality
technical workforce we must confront this issue. We need to stop the rush to
calculus and focus instead on a thorough grounding in algebra, geometry and
functions.
Calculus is one of the great intellectual achievements of the last 400 years
; shortchanging it by reducing its beauty and utility to a list of problems
to be checked off a rubric does a disservice to everyone. | J*****3
发帖数: 25 | | l********1
发帖数: 1 | 3 我老早就说了，因式分解，整式运算都吭哧憋肚的， 如何学习微积分？
我这的学生，你吗GRF都算的替他糊涂， 你能相信人家是微积分AP。
美国这种功利社会， 教育就把完蛋了
【在 l********1 的大作中提到】
: The rush to calculus is bad for students and their futures in STEM
: November 20, 2015 6.12am EST
: Author
: Kevin Knudson
: Professor of Mathematics, University of Florida
: Partners
: University of Florida
: University of Florida provides funding as a founding partner of The
: Conversation US.
: View all partners
| F*****d
发帖数: 2848 | | V*******n
发帖数: 1 | | w***u
发帖数: 17713 | 6 对啊，还不如在AP微积分和物理之前开一门近代数学物理史，让学生明白这些东西的来
龙去脉的逻辑。 | l********1
发帖数: 1 | 7 没错。
让学生理解人类科学思想的进化过程
【在 w***u 的大作中提到】
: 对啊，还不如在AP微积分和物理之前开一门近代数学物理史，让学生明白这些东西的来
: 龙去脉的逻辑。
| f*******y
发帖数: 8358 | 8 完全正确。高中ap课就是毒瘤。
这帮小孩子自以为自己会了，其实屁都不会。
一到工程运用，全都傻眼。 |
|
