Calculating Beyond Their Years
While we complain about the low performance of American Students in international math comparisons, while we whine that educational standards continually decline, some high school students are getting more and more advanced math educations.
Are the detractors right? Are we losing "rich variety," breadth, and depth in the curriculum in order to move students further along? Are we advancing students too soon into topics that they are not mature enough to handle?
Ironically, at the same time, colleges are burdened with ever more remedial courses in math (as well as reading and writing) for those students who did not learn even the basics sufficiently for an advanced education. We (the colleges) teach ninth grade algebra, or even 7-8th grade arithmetic, while the high schools teach second-year college math.
You know what I think? We should do away with the rigid 4-year notion about high school. We should define a level of knowledge as a target (something like the GED), and whenever students reach that level they should get their diploma and move on to college. If the state wants to pay for college courses until the age of 18, fine.
Of course, there is a drawback to that idea, and it is this: I am making the assumption that colleges do a better job of teaching college-level material than high schools do. This is sometimes the case, but not always. An experienced high school teacher, even one who has to go back to 30-year old notes, might be more effective at teaching Calc III than an adjunct college instructor with a fresh Master's degree, or a PhD professor whose heart is in research and who teaches only because he has to. However, college teaching horror stories might not be as common as they are popular in the retelling, and the corresponding horror stories about high school teachers who do not know enough and end up instilling students with incorrect concepts are far less likely to be heard.
In fact, college teachers routinely complain about students who have had calculus in high school, but learned it inadequately, either with too little emphasis on concepts (they memorized skills and tricks but never did proofs and cannot explain the meaning of their work) or with too much reliance on calculators, so that basic skills in calculation are underdeveloped.
What about the loss of "rich variety" in topics like geometry and trigonometry? Without knowing the actual curriculum being used, I can't really make a fair comment about the schools in question. However, even before the "calculus push" there was a great deal lost in these topics. Sometime during the 1970's and 80's, high school geometry and trigonometry were gutted and refurbished...and indeed one could question whether the changes have resulted in a richer curriculum or a poorer one. Mathematics has taken a hard hit from the near-removal of proof from the high school curriculum. The new topics that have been added may not be as important or beneficial. It seems clear to me that the changes were made to make math more "accessible" to students (i.e., dumbed down) as we moved to requiring more math for all students. In recent years, there seems to be a trend toward reclaiming some of the lost ground, and courses like "pre-calculus" may, in some cases, be as challenging and comprehensive as anything from the past. But I have seen no evidence that logic and proof have made a comeback in the high school curriculum. Without these, we cannot have a truly "advanced" high school mathematics program.
Calculating Beyond Their Years
For More High School Students, AP Math Just Isn't Sufficient
By Daniel de Vise
Washington Post Staff Writer
Monday, February 6, 2006; B01
More than 500 students in the Montgomery and Fairfax school systems...are taking multivariable calculus, a course traditionally taken by math majors in their second year of college....
Driving the trend is a conviction that algebra...should be taught as soon as students are ready to learn it. Students with a flicker of math talent are taking the high school Algebra I course in eighth grade, if not before.
When the school tapped Moriarty to teach the new course last year, he had to search his basement to find notes he took on multivariable calculus as a sophomore in college almost 30 years ago. "So, we're in the learning stages, and I'm learning, as well as the kids," he said.
Possibly lost along the way: the rich variety of topics in algebra, plane and solid geometry, trigonometry and mathematical applications that once occupied students through much of their journey through high school. "Kids are missing out on some very important concepts that have been pushed aside to make room for calculus," said Alfred S. Posamentier, dean of the School of Education at the City College of New York....
"We got it," a group of girls boasted from the back of the class. One of them, 17-year-old Alice Chen, smiled and said, "I actually understand this."
Are the detractors right? Are we losing "rich variety," breadth, and depth in the curriculum in order to move students further along? Are we advancing students too soon into topics that they are not mature enough to handle?
Ironically, at the same time, colleges are burdened with ever more remedial courses in math (as well as reading and writing) for those students who did not learn even the basics sufficiently for an advanced education. We (the colleges) teach ninth grade algebra, or even 7-8th grade arithmetic, while the high schools teach second-year college math.
You know what I think? We should do away with the rigid 4-year notion about high school. We should define a level of knowledge as a target (something like the GED), and whenever students reach that level they should get their diploma and move on to college. If the state wants to pay for college courses until the age of 18, fine.
Of course, there is a drawback to that idea, and it is this: I am making the assumption that colleges do a better job of teaching college-level material than high schools do. This is sometimes the case, but not always. An experienced high school teacher, even one who has to go back to 30-year old notes, might be more effective at teaching Calc III than an adjunct college instructor with a fresh Master's degree, or a PhD professor whose heart is in research and who teaches only because he has to. However, college teaching horror stories might not be as common as they are popular in the retelling, and the corresponding horror stories about high school teachers who do not know enough and end up instilling students with incorrect concepts are far less likely to be heard.
In fact, college teachers routinely complain about students who have had calculus in high school, but learned it inadequately, either with too little emphasis on concepts (they memorized skills and tricks but never did proofs and cannot explain the meaning of their work) or with too much reliance on calculators, so that basic skills in calculation are underdeveloped.
What about the loss of "rich variety" in topics like geometry and trigonometry? Without knowing the actual curriculum being used, I can't really make a fair comment about the schools in question. However, even before the "calculus push" there was a great deal lost in these topics. Sometime during the 1970's and 80's, high school geometry and trigonometry were gutted and refurbished...and indeed one could question whether the changes have resulted in a richer curriculum or a poorer one. Mathematics has taken a hard hit from the near-removal of proof from the high school curriculum. The new topics that have been added may not be as important or beneficial. It seems clear to me that the changes were made to make math more "accessible" to students (i.e., dumbed down) as we moved to requiring more math for all students. In recent years, there seems to be a trend toward reclaiming some of the lost ground, and courses like "pre-calculus" may, in some cases, be as challenging and comprehensive as anything from the past. But I have seen no evidence that logic and proof have made a comeback in the high school curriculum. Without these, we cannot have a truly "advanced" high school mathematics program.
posted by Dr. Stat | Monday, February 13, 2006
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