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The following remarks were written by me to my sons' high school community some years ago. I believe they are still relevant. They were then considered so heretical that they were not acceptable as publishable.
The idea of posting these remarks was inspired by Zapper Z's extensive and helpful essays on preparation for future physicists, and the response from students requesting guidance for preparing for mathematics and that at the high school level.
Some remarks on high school preparation for a college education:
Some time ago I argued that XXX could not be considered a particularly "hard" school in comparison with many others, because it had so few AP courses. Now that this is changing, I have begun to have some reservations. I had hoped AP courses would strengthen the program by upgrading weaker course offerings, rather than competing against the excellent courses already in place. I believe that in the country as a whole, this may have been a principal result of the proliferation of AP courses, tending to a sort of standardization of advanced instruction, bringing a reduction in quality of education at good high schools, rather than an overall upgrading of the level of the average course offering. I also did not realize that graduates of "Advanced Placement" courses would take the term too literally and try to place out of substantive courses in college which they should have taken. In the case of those AP students who repeat beginning college courses I have also found the problem of trying to teach in depth a college subject to people who think they have already learned it.
I am most familiar with mathematics, which I teach at the University of Georgia, so I use that example. The AP designation in calculus refers to a specific list of "topics" on which one must be prepared to work problems. A year of this AP material coincides with the content of one or two quarters of non honors college calculus at Georgia, but a full year college course, and especially an honors course, not only covers more ground, but treats the material at greater depth. It is ironic that AP calculus courses, which are taken by honors high school students, are comparable at best to non honors college courses, which the best such students would not elect.
As a result many entering AP college students place either into advanced, but less stimulating, non honors courses, or into intermediate honors college mathematics courses for which they are not prepared. Before the AP revolution, students prepared by getting a better grounding in algebra and geometry (and sometimes logic) than is found in high schools today, then took a first year college calculus course which included theory. Introductory college calculus courses for gifted mathematics students which teach theory as well as computation are hard to find today because so many students exempt this course with AP credit. The disappearance of the most outstanding introductory college calculus courses is thus a direct result of the proliferation of significantly inferior AP courses.
In view of its unsuitability, it is ironic that AP credit has begun to be designated as the "prerequisite" to some advanced courses, even though the true prerequisite for advanced work is often just the ability to think in a certain way. This may be the case even when the college catalog says otherwise. At Stanford for example, the prerequisite listed recently for honors intermediate calculus is a certain score on the AP calculus exam, but when asked, the departmental advisor said "Of course that's not the real prerequisite" (his emphasis). The real prerequisite? "To be able to handle proofs, with no apology". The book used in that course is volume 2 of Apostol, an outstanding text treating calculus with theory. Presumably the right preparation is to learn beginning calculus from volume 1 of Apostol, but where can the interested student find such a course? Stanford does not offer it (that's the course that was replaced by the AP courses), and it certainly is not available in most high school AP classes; (books used in the XXX course are ordinarily one or even two levels of sophistication below Apostol). The result of this at Stanford is roughly a 70% attrition rate (after the first week!) in the honors intermediate calculus class, among those students who have the required score on the AP test. Surely many of those students who must drop out are disappointed that they are not in fact prepared for the course, and possibly the career, they had wanted.
Unfortunately AP calculus courses and the standardized testing mentality have helped to eliminate, not just from the college Freshman mathematics curriculum but also from high schools, classes in which theory and proof (i.e. systematic logical reasoning) are taught, since "proofs" are seldom included on AP tests. For example the 1982 and 1987 AP BC calculus tests in my practice book have less than 3% proof questions, whereas the exams in the Stanford course above are said to be100% proofs. This phenomenon has accompanied years of decline, and the current near extinction, of adequate teaching of geometry in high school, which worsens the problem of learning either calculus or deductive reasoning.
I conjecture that these negative effects are not so great in some subjects where AP exemption is less common. For instance my impression is that in the recent past students from XXX's non AP honors English courses have been superbly prepared for beginning college courses in that subject. Presumably the reason is that in these classes, students learn to read, write, and discuss their ideas. I hope these courses are never replaced by ones designed to prepare people to answer multiple choice questions on the correct author of some obscure poem.
Interestingly, although our data at the University of Georgia shows no correlation (and even some negative correlation!), between scores on the quantitative SAT test and performance in our precalculus and basic non honors calculus courses, there does apparently exist a positive correlation (almost a direct one) with scores on the verbal SAT test. My own theory is that the verbal test, as bad as it is, at least measures vocabulary (mathematics is a language), and the ability to comprehend what one has read. Consequently the demise not only of instruction in reasoning in mathematics, but the decline in the ability of the average student to read and write, has steadily tracked the drop in performance also in basic mathematics courses for non honors students. (One might even argue from this that the claim that Saxon's books raise SATQ test scores, also suggests that they may lower average performance in college mathematics courses.)
What is my conclusion? I suggest the school seriously reconsider the practice of creating AP courses in subjects which are already represented by excellent honors courses, since this may well lead to the demise of the superior course, and a decrease in the quality of student preparation. In subjects where AP courses already compete with non AP courses, I strongly urge students to select the course which involves the most writing, and the deepest analysis, without regard to which one boasts a syllabus sanctioned by the Educational Testing Service. In cases where an AP course has already driven a superior course out of existence, I feel there is a strong argument for creating, or recreating, a non AP honors alternative.
I do not oppose taking AP classes in principle, but since (in my experience) they do not play an appropriate role in advanced college placement, I do believe they must justify themselves based simply on their educational merits. I also strongly suggest that a graduate of an AP class consider taking an introductory college honors course in the same subject rather than skipping the introductory course altogether.
The only case in which I see a reason to consider creating an AP class is in a subject where the existing course work is currently on an inappropriately low level. Even in such cases I think it likely that a non AP honors course designed by the teacher may be an even better option. In my opinion such an opportunity exists at XXX in the physics program, which I understand does not ordinarily offer a calculus based course. One possible way to make good use of the existing AP calculus course would be to offer a subsequent or concurrent calculus based physics course, or even a course that combined the two subjects. Since Newton invented calculus precisely to do physics, this is one of the best possible ways to learn both physics and calculus.
From my own perspective I believe there is also a real need for new substantive mathematics courses which are not just oriented towards performance on standardized tests. When I tell my colleagues at the University of Georgia that XXX does not offer a year long course in geometry for example, they do not readily believe me. I would also like to see innovative, faculty - sponsored, courses on other subjects of current or abiding importance in mathematics and related areas, such as linear algebra (an easier and more fundamental subject than calculus), finite mathematics and probabilty, computer programming, algorithms, numerical analysis, or computer aided design.
In general, I believe those of us who are "consumers" of XXX educations, parents and students, should have faith in the knowledge and scholarship of the teachers; these outstanding individuals should be considered at least as qualified to select the content of their courses as the faceless people who write standardized tests. Some of these teachers value and use an AP syllabus in their own courses, which is a recommendation to me of the positive aspects of some AP programs. Others prefer to design their own curricula. Such distinctive courses offer opportunities unique to XXX, and I believe they play a large role in the school's impressively successful identity. Some teachers may even be holding back exciting proposals thinking we want only standardized education from them. I hope such individually conceived courses will continue to be encouraged, and valued for the rare gems that they are.
For the students who must enroll in the courses if they are to survive, I suggest you remember primarily to try to educate yourself. In particular try not to let the quest for a flawless GPA prevent you from studying subjects you find difficult. Even if science courses are hard for you, how much can you understand about our world if you don't know at least something of biology, chemistry, physics, and (yes) mathematics? If you would enjoy going to Paris, or Madrid, it would help to speak French or Spanish. If you think art and music classes are not valuable, you might think about how you are going to create a beautiful environment in your apartment or home, or your life, without such knowledge.
Now what about the "real world" of getting into college or getting a job? Is it practical to just go along learning to read, write, reflect, analyze, discuss, and play, when you fear that college admissions officials are going to judge you based mainly on your standardized test profile? May I respectfully suggest we all try not to hyperventilate over college admission. From my own experience as a college professor, and reader and writer of countless recommendation letters, I recommend to you to be curious, to be diligent, and to pursue activities for which you have real enthusiasm. If you have a genuine enjoyment for learning, if you have thought deeply about any significant topic, if you have worked hard to accomplish something in any area, if you can express yourself well and have practiced discussing your ideas with others, it will come through in your college essay or interview as well as in your letters of recommendation.
I believe too, the tight job market in higher education over the last couple of decades means more and more colleges are now assembling the most qualified faculties they have ever had. Certainly this is true in mathematics. If you honestly embrace your XXX education, I believe you are virtually assured of admission to a college which offers more than anyone person can possibly absorb. This is borne out by a glance at recent lists of admissions, and by speaking with recent graduates. Since there is a shortage of well qualified students at most colleges, there may be even a slight danger that you will get into a school which is actually too challenging.
And if after all you find yourself in a situation where you seem to need credentials you don't have? A positive attitude always helps overcome gaps in your vita. I have often been inspired by a story my mother told me about her interview for a secretarial job she needed badly during the great depression. When asked if she had any experience, she said "No, but I can learn to do anything anybody else can do." She got the job. You can too.
Roy Smith
The idea of posting these remarks was inspired by Zapper Z's extensive and helpful essays on preparation for future physicists, and the response from students requesting guidance for preparing for mathematics and that at the high school level.
Some remarks on high school preparation for a college education:
Some time ago I argued that XXX could not be considered a particularly "hard" school in comparison with many others, because it had so few AP courses. Now that this is changing, I have begun to have some reservations. I had hoped AP courses would strengthen the program by upgrading weaker course offerings, rather than competing against the excellent courses already in place. I believe that in the country as a whole, this may have been a principal result of the proliferation of AP courses, tending to a sort of standardization of advanced instruction, bringing a reduction in quality of education at good high schools, rather than an overall upgrading of the level of the average course offering. I also did not realize that graduates of "Advanced Placement" courses would take the term too literally and try to place out of substantive courses in college which they should have taken. In the case of those AP students who repeat beginning college courses I have also found the problem of trying to teach in depth a college subject to people who think they have already learned it.
I am most familiar with mathematics, which I teach at the University of Georgia, so I use that example. The AP designation in calculus refers to a specific list of "topics" on which one must be prepared to work problems. A year of this AP material coincides with the content of one or two quarters of non honors college calculus at Georgia, but a full year college course, and especially an honors course, not only covers more ground, but treats the material at greater depth. It is ironic that AP calculus courses, which are taken by honors high school students, are comparable at best to non honors college courses, which the best such students would not elect.
As a result many entering AP college students place either into advanced, but less stimulating, non honors courses, or into intermediate honors college mathematics courses for which they are not prepared. Before the AP revolution, students prepared by getting a better grounding in algebra and geometry (and sometimes logic) than is found in high schools today, then took a first year college calculus course which included theory. Introductory college calculus courses for gifted mathematics students which teach theory as well as computation are hard to find today because so many students exempt this course with AP credit. The disappearance of the most outstanding introductory college calculus courses is thus a direct result of the proliferation of significantly inferior AP courses.
In view of its unsuitability, it is ironic that AP credit has begun to be designated as the "prerequisite" to some advanced courses, even though the true prerequisite for advanced work is often just the ability to think in a certain way. This may be the case even when the college catalog says otherwise. At Stanford for example, the prerequisite listed recently for honors intermediate calculus is a certain score on the AP calculus exam, but when asked, the departmental advisor said "Of course that's not the real prerequisite" (his emphasis). The real prerequisite? "To be able to handle proofs, with no apology". The book used in that course is volume 2 of Apostol, an outstanding text treating calculus with theory. Presumably the right preparation is to learn beginning calculus from volume 1 of Apostol, but where can the interested student find such a course? Stanford does not offer it (that's the course that was replaced by the AP courses), and it certainly is not available in most high school AP classes; (books used in the XXX course are ordinarily one or even two levels of sophistication below Apostol). The result of this at Stanford is roughly a 70% attrition rate (after the first week!) in the honors intermediate calculus class, among those students who have the required score on the AP test. Surely many of those students who must drop out are disappointed that they are not in fact prepared for the course, and possibly the career, they had wanted.
Unfortunately AP calculus courses and the standardized testing mentality have helped to eliminate, not just from the college Freshman mathematics curriculum but also from high schools, classes in which theory and proof (i.e. systematic logical reasoning) are taught, since "proofs" are seldom included on AP tests. For example the 1982 and 1987 AP BC calculus tests in my practice book have less than 3% proof questions, whereas the exams in the Stanford course above are said to be100% proofs. This phenomenon has accompanied years of decline, and the current near extinction, of adequate teaching of geometry in high school, which worsens the problem of learning either calculus or deductive reasoning.
I conjecture that these negative effects are not so great in some subjects where AP exemption is less common. For instance my impression is that in the recent past students from XXX's non AP honors English courses have been superbly prepared for beginning college courses in that subject. Presumably the reason is that in these classes, students learn to read, write, and discuss their ideas. I hope these courses are never replaced by ones designed to prepare people to answer multiple choice questions on the correct author of some obscure poem.
Interestingly, although our data at the University of Georgia shows no correlation (and even some negative correlation!), between scores on the quantitative SAT test and performance in our precalculus and basic non honors calculus courses, there does apparently exist a positive correlation (almost a direct one) with scores on the verbal SAT test. My own theory is that the verbal test, as bad as it is, at least measures vocabulary (mathematics is a language), and the ability to comprehend what one has read. Consequently the demise not only of instruction in reasoning in mathematics, but the decline in the ability of the average student to read and write, has steadily tracked the drop in performance also in basic mathematics courses for non honors students. (One might even argue from this that the claim that Saxon's books raise SATQ test scores, also suggests that they may lower average performance in college mathematics courses.)
What is my conclusion? I suggest the school seriously reconsider the practice of creating AP courses in subjects which are already represented by excellent honors courses, since this may well lead to the demise of the superior course, and a decrease in the quality of student preparation. In subjects where AP courses already compete with non AP courses, I strongly urge students to select the course which involves the most writing, and the deepest analysis, without regard to which one boasts a syllabus sanctioned by the Educational Testing Service. In cases where an AP course has already driven a superior course out of existence, I feel there is a strong argument for creating, or recreating, a non AP honors alternative.
I do not oppose taking AP classes in principle, but since (in my experience) they do not play an appropriate role in advanced college placement, I do believe they must justify themselves based simply on their educational merits. I also strongly suggest that a graduate of an AP class consider taking an introductory college honors course in the same subject rather than skipping the introductory course altogether.
The only case in which I see a reason to consider creating an AP class is in a subject where the existing course work is currently on an inappropriately low level. Even in such cases I think it likely that a non AP honors course designed by the teacher may be an even better option. In my opinion such an opportunity exists at XXX in the physics program, which I understand does not ordinarily offer a calculus based course. One possible way to make good use of the existing AP calculus course would be to offer a subsequent or concurrent calculus based physics course, or even a course that combined the two subjects. Since Newton invented calculus precisely to do physics, this is one of the best possible ways to learn both physics and calculus.
From my own perspective I believe there is also a real need for new substantive mathematics courses which are not just oriented towards performance on standardized tests. When I tell my colleagues at the University of Georgia that XXX does not offer a year long course in geometry for example, they do not readily believe me. I would also like to see innovative, faculty - sponsored, courses on other subjects of current or abiding importance in mathematics and related areas, such as linear algebra (an easier and more fundamental subject than calculus), finite mathematics and probabilty, computer programming, algorithms, numerical analysis, or computer aided design.
In general, I believe those of us who are "consumers" of XXX educations, parents and students, should have faith in the knowledge and scholarship of the teachers; these outstanding individuals should be considered at least as qualified to select the content of their courses as the faceless people who write standardized tests. Some of these teachers value and use an AP syllabus in their own courses, which is a recommendation to me of the positive aspects of some AP programs. Others prefer to design their own curricula. Such distinctive courses offer opportunities unique to XXX, and I believe they play a large role in the school's impressively successful identity. Some teachers may even be holding back exciting proposals thinking we want only standardized education from them. I hope such individually conceived courses will continue to be encouraged, and valued for the rare gems that they are.
For the students who must enroll in the courses if they are to survive, I suggest you remember primarily to try to educate yourself. In particular try not to let the quest for a flawless GPA prevent you from studying subjects you find difficult. Even if science courses are hard for you, how much can you understand about our world if you don't know at least something of biology, chemistry, physics, and (yes) mathematics? If you would enjoy going to Paris, or Madrid, it would help to speak French or Spanish. If you think art and music classes are not valuable, you might think about how you are going to create a beautiful environment in your apartment or home, or your life, without such knowledge.
Now what about the "real world" of getting into college or getting a job? Is it practical to just go along learning to read, write, reflect, analyze, discuss, and play, when you fear that college admissions officials are going to judge you based mainly on your standardized test profile? May I respectfully suggest we all try not to hyperventilate over college admission. From my own experience as a college professor, and reader and writer of countless recommendation letters, I recommend to you to be curious, to be diligent, and to pursue activities for which you have real enthusiasm. If you have a genuine enjoyment for learning, if you have thought deeply about any significant topic, if you have worked hard to accomplish something in any area, if you can express yourself well and have practiced discussing your ideas with others, it will come through in your college essay or interview as well as in your letters of recommendation.
I believe too, the tight job market in higher education over the last couple of decades means more and more colleges are now assembling the most qualified faculties they have ever had. Certainly this is true in mathematics. If you honestly embrace your XXX education, I believe you are virtually assured of admission to a college which offers more than anyone person can possibly absorb. This is borne out by a glance at recent lists of admissions, and by speaking with recent graduates. Since there is a shortage of well qualified students at most colleges, there may be even a slight danger that you will get into a school which is actually too challenging.
And if after all you find yourself in a situation where you seem to need credentials you don't have? A positive attitude always helps overcome gaps in your vita. I have often been inspired by a story my mother told me about her interview for a secretarial job she needed badly during the great depression. When asked if she had any experience, she said "No, but I can learn to do anything anybody else can do." She got the job. You can too.
Roy Smith
