[math-fun] High school calculus?
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing. Rich ------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are: (1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening. (2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA. (3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization. Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well. The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum? Cheers! Amy
I'm sympathetic to these viewpoints. One goal might be to craft a proposal for a high school Discrete Math AP exam. Here is the document that to a large extent controls the USA high school math curriculum: http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-d... For this imagined Discrete Math alternative, some juicier name might be better...just brainstorming AP Math for the Knowledge Economy (yuck) AP Computational Math AP Math for the Computational Sciences Here is a list of all the AP exams http://apcentral.collegeboard.com/apc/public/courses/descriptions/index.html I don't know how/when certain tests were added (for example, Macroeconomics), but however they got that done, that is the work to be done here. High schools will teach it if there is an AP exam. On Fri, Mar 11, 2011 at 3:59 PM, <rcs@xmission.com> wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
John G. Kemeny, a Dartmouth College mathematics professor, dealt with all of these issues first hand in the 1960's and learned a lot in the process. The questions you ask led him and his colleague Thomas Kurtz to transform the entire institution during the 1960's and 1970's, well ahead of the trend. The following quotes are from "The Computer and the Campus: an Interview of John Kemeny" (an interview by Ken King of EDUCOM performed in 1991, a year before Kemeny's death). They address your questions very closely, and the rest of the interview is relevant too. This excerpt starts at about 13:05: *KK: How do you deal with [resistance of faculty to resist tradition by adding computing to the curriculum]?* * * *JGK: I think the question that would have to be asked of the administrators at the institution is: "How would you evaluate a faculty member who teaches a completely obsolete course superbly?" I would think, I would *hope* they would give low grades. And in certain types of courses, teaching it without using computers is teaching a completely obsolete course.* And this is at 14:10: *JGK: In a calculus course, Taylor series is a fairly difficult topic. The thought that you add up a bunch of polynomials, which are terribly nice functions, the sum of them will be very close to some very weird function that you started with, the transcendental function, is very difficult for a student to understand.* * I followed the mathematics in college, but I had absolutely no feeling deep down about what was going on. On a computer, you can have the original function drawn, and show step by step as you add up the polynomials, you draw the graphs, and you see it getting closer and closer to the original function.* * I learned things about Taylor series I had not known after twenty years of teaching calculus just by looking at the graphs that appeared on the computer screen.* The entire interview is here: http://www.dartmouth.edu/comp/about/archive/history/kemeny/ or on Youtube, here: http://www.youtube.com/v/HHi3VFOL-AI - Robert
From: Anna Johnston [jannaston@gmail.com]
Subject: math fun and calculus
Hi Rich,
I wanted to ask you a question (and perhaps get a gage through mathfun) --
old though it may be -- that's come up in my current circle: calculus in
high school. I've read MAA articles dating back 11 years on the
questionable nature of teaching calculus in high school, talked to high
school teachers, mathematician parents, as well as my former colleagues at
WSU. The basic thread of these conversations is that calculus should not be
part of the high school curriculum. Instead there should be more breadth
with a stronger emphasis on discrete concepts (combinatorics, number theory,
probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help
students think logically while giving them knowledge far more useful in
everyday life. [...]
(2) Secondly, society's need for calculus has been surpassed by the need for
discrete math. [...]
(3) Thirdly, the linear push to calculus is a turn off to many students.
The style of teaching most HS calculus teachers are forced into teaches
students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and
stats. [...]
The question is: Why is calculus still the perceived linear end point to HS
math and what is the best way to change perceptions and curriculum?
-- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com
I'd like to see more geometry (and topology) as well. I'm still not clear why the Eastern European emphasis on geometry never caught on in the US despite some traction eg George Pólya. On Fri, Mar 11, 2011 at 9:27 PM, Robert Munafo <mrob27@gmail.com> wrote:
John G. Kemeny, a Dartmouth College mathematics professor, dealt with all of these issues first hand in the 1960's and learned a lot in the process. The questions you ask led him and his colleague Thomas Kurtz to transform the entire institution during the 1960's and 1970's, well ahead of the trend.
The following quotes are from "The Computer and the Campus: an Interview of John Kemeny" (an interview by Ken King of EDUCOM performed in 1991, a year before Kemeny's death). They address your questions very closely, and the rest of the interview is relevant too.
This excerpt starts at about 13:05:
*KK: How do you deal with [resistance of faculty to resist tradition by adding computing to the curriculum]?* * * *JGK: I think the question that would have to be asked of the administrators at the institution is: "How would you evaluate a faculty member who teaches a completely obsolete course superbly?" I would think, I would *hope* they would give low grades. And in certain types of courses, teaching it without using computers is teaching a completely obsolete course.*
And this is at 14:10:
*JGK: In a calculus course, Taylor series is a fairly difficult topic. The thought that you add up a bunch of polynomials, which are terribly nice functions, the sum of them will be very close to some very weird function that you started with, the transcendental function, is very difficult for a student to understand.* * I followed the mathematics in college, but I had absolutely no feeling deep down about what was going on. On a computer, you can have the original function drawn, and show step by step as you add up the polynomials, you draw the graphs, and you see it getting closer and closer to the original function.* * I learned things about Taylor series I had not known after twenty years of teaching calculus just by looking at the graphs that appeared on the computer screen.*
The entire interview is here:
http://www.dartmouth.edu/comp/about/archive/history/kemeny/
or on Youtube, here:
http://www.youtube.com/v/HHi3VFOL-AI
- Robert
From: Anna Johnston [jannaston@gmail.com]
Subject: math fun and calculus
Hi Rich,
I wanted to ask you a question (and perhaps get a gage through mathfun) --
old though it may be -- that's come up in my current circle: calculus in
high school. I've read MAA articles dating back 11 years on the
questionable nature of teaching calculus in high school, talked to high
school teachers, mathematician parents, as well as my former colleagues at
WSU. The basic thread of these conversations is that calculus should not be
part of the high school curriculum. Instead there should be more breadth
with a stronger emphasis on discrete concepts (combinatorics, number theory,
probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help
students think logically while giving them knowledge far more useful in
everyday life. [...]
(2) Secondly, society's need for calculus has been surpassed by the need for
discrete math. [...]
(3) Thirdly, the linear push to calculus is a turn off to many students.
The style of teaching most HS calculus teachers are forced into teaches
students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and
stats. [...]
The question is: Why is calculus still the perceived linear end point to HS
math and what is the best way to change perceptions and curriculum?
-- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Another issue with the mathematics curriculum in the United Kingdom is the lack of Euclidean geometry. Throughout the entire GCSE and A-level syllabus (equivalent to 'high school'), there is nothing beyond Pythagoras' theorem, basic trigonometry, and Thales' theorem. The triangle centres are never even suggested to students (except the centroid as the 'centre of mass' during Mechanics modules), and there is no mention of Ceva's theorem. Discrete maths are taught in the A-level syllabus, but only in the most laborious way possible. The majority of the course involves weighted graphs and optimisation, followed by a bombardment of algorithms capable of dissuading all but the most tenacious of students. Rather than devising algorithms (i.e. programming), the emphasis is on mechanically following algorithms without any insight as to why the algorithms work. The course (entitled 'decision maths') reduces people to mindless automata. Believe it or not, the Decision Maths course even involves *solving the Travelling Salesman Problem* -- how torturous is that?! It could be replaced with something much more enjoyable, such as Fourier transforms. In my opinion, concepts such as binary and Turing machines should be taught in school, mainly as preparation for informatics/computer science. What are your views on this? Sincerely, Adam P. Goucher
IMHO the problems in the UK go back to the choices of course material when the switch to all comprehensive schools was made. Those who supported switching to all comprehensive won the argument by saying that the more academic students would have a positive effect on the less academic and not vice-versa. Well I would have agreed with that *except* before the switch to comprehensive in general the Grammar school syllabus (in nearly all subjects) was more advanced than the Secondary Modern syllabus and what did they do ? They switched all over to the lesser syllabus thus completely negating any advantage created by the switch to comprehensive. IMHO the answer is proper streaming in *all* subjects and at least 2 different syllabuses in each subject depending on the ability/aptitude of the student in that subject, in fact preferably more than 2. When I was at Grammar School (30+ years ago) we were streamed into 5 sets (each of around 32 pupils) for maths and even then sets 1 to 4 did the "O" level syllabus (with calculus) but set 5 did the lesser CSE (no calculus). In a Comprehensive situation for the lower sets math study should in fact concentrate on everyday maths e.g. the maths required to maintain good family finances etc. and nothing much more than this - I say this as it should be remembered that in the Grammar school system as I described even set 5 at the Grammar school were apparently part of the top 45% of the population academically speaking. On 12 Mar 2011, at 11:43, Adam P. Goucher wrote:
Another issue with the mathematics curriculum in the United Kingdom is the lack of Euclidean geometry. Throughout the entire GCSE and A-level syllabus (equivalent to 'high school'), there is nothing beyond Pythagoras' theorem, basic trigonometry, and Thales' theorem. The triangle centres are never even suggested to students (except the centroid as the 'centre of mass' during Mechanics modules), and there is no mention of Ceva's theorem.
Discrete maths are taught in the A-level syllabus, but only in the most laborious way possible. The majority of the course involves weighted graphs and optimisation, followed by a bombardment of algorithms capable of dissuading all but the most tenacious of students.
Rather than devising algorithms (i.e. programming), the emphasis is on mechanically following algorithms without any insight as to why the algorithms work. The course (entitled 'decision maths') reduces people to mindless automata.
Believe it or not, the Decision Maths course even involves *solving the Travelling Salesman Problem* -- how torturous is that?! It could be replaced with something much more enjoyable, such as Fourier transforms.
In my opinion, concepts such as binary and Turing machines should be taught in school, mainly as preparation for informatics/computer science.
What are your views on this?
Sincerely,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Re _not_ teaching calculus in high school: I have a great deal of sympathy for this view. Some background: I was a member of our high school's math team that placed highly in the state of Ohio (first, I think, but my memory about this event isn't very good), and I won the Cincinnati math contest my senior year. My senior year, I took all AP courses (except English!), including AP math, which was calculus, more-or-less. As a result of these courses, I was able to "place" out of some courses my freshman year at MIT, including 18.01. My oldest sister's boyfriend was a college math&physics major (who later got his physics doctorate from Heidelberg), and he tried to get me to understand some of the subtleties of "epsilon-delta" calculus, but I have to say that I was not ready for it, and I didn't really understand it. In retrospect, I should have stopped wasting all of my high school teachers' time and gone straight to MIT after my junior year in high school. Even though my high school had some of the best high school teachers in the U.S., they still were a far cry from what MIT had to offer. When I arrived at MIT & started learning _real_ math & _real_ physics, I felt like I had gone to heaven & I knew that I had just been wasting time in high school. Calculus never excited me much in high school (or MIT, for that matter). I was far more interested in algebra & mathematical logic. For me, the coolest thing about calculus was the algebraic operations of derivation and antiderivation. I never really "understood" infinitesimal calculus until I took probability theory and especially advanced mathematical logic, where I got to understand models and cardinal numbers and different sizes of infinities. I took an extracurricular number theory course while I was still in high school and _loved_ it. Later, at MIT, I took an algebra course where we studied groups & rings & fields (including finite fields for coding theory), and was really turned on by the beauty of these constructs. I also really loved my undergraduate combinatorics course -- especially using generating functions & counting theory. My bible for learning a lot of math that I never had in courses has been Knuth's 3-volume set. I think that if this set had been available when I was still in high school, I would have gone through and done every exercise on my own. I could easily see giving up calculus in high school and learning more algebra -- including how to use a computer symbolic algebra system -- and more geometry in the form of computer graphics. I never understood why mathematicians of the middle 20th century wanted to wring all of the pleasure out of math by teaching everything so abstractly; a healthy dose of actual number crunching (aided by a computer, of course) is a much better preparation for calculus. (Perhaps I'm too fond of 19th century mathematics.) Also, matrix algebra -- taught correctly -- can be quite beautiful in its own right, and with excellent computer software available, it should be accessible to high school students. At 03:59 PM 3/11/2011, rcs@xmission.com wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
Here Calculus has apparently been removed from the National under-16 syllabus which I find a little annoying (especially when successive Governments have claimed there's been no "dumbing down" of education in order to improve pass rates) *because calculus hasn't been replaced by something else*. IMHO the whole of education needs a complete overhaul - it should be designed so the same knowledge is acquired but with 100* the fun factor from the students perspective - to me this means rethink the teaching method completely so it's all done through games and challenges for the students rather than dry lecturing/dictation/textbook use etc. IMHO the real key being a massive increase in *expectations* for all years from 6 to 11. Plus get rid of the incredibly stupid and crippling modern "PC" attitude of removing all sense of competition as if it's morally or ethically incorrect - just congratulate the "winners" and commiserate with the "losers", to do otherwise teachers children something that simply is a lie and some will be unprepared for the naked truth once they leave education. (I realise that this 'PC" practice only applies to some and not all). On 12 Mar 2011, at 00:49, Henry Baker wrote:
Re _not_ teaching calculus in high school:
I have a great deal of sympathy for this view.
Some background: I was a member of our high school's math team that placed highly in the state of Ohio (first, I think, but my memory about this event isn't very good), and I won the Cincinnati math contest my senior year. My senior year, I took all AP courses (except English!), including AP math, which was calculus, more-or-less. As a result of these courses, I was able to "place" out of some courses my freshman year at MIT, including 18.01. My oldest sister's boyfriend was a college math&physics major (who later got his physics doctorate from Heidelberg), and he tried to get me to understand some of the subtleties of "epsilon-delta" calculus, but I have to say that I was not ready for it, and I didn't really understand it.
In retrospect, I should have stopped wasting all of my high school teachers' time and gone straight to MIT after my junior year in high school. Even though my high school had some of the best high school teachers in the U.S., they still were a far cry from what MIT had to offer. When I arrived at MIT & started learning _real_ math & _real_ physics, I felt like I had gone to heaven & I knew that I had just been wasting time in high school.
Calculus never excited me much in high school (or MIT, for that matter). I was far more interested in algebra & mathematical logic. For me, the coolest thing about calculus was the algebraic operations of derivation and antiderivation. I never really "understood" infinitesimal calculus until I took probability theory and especially advanced mathematical logic, where I got to understand models and cardinal numbers and different sizes of infinities. I took an extracurricular number theory course while I was still in high school and _loved_ it. Later, at MIT, I took an algebra course where we studied groups & rings & fields (including finite fields for coding theory), and was really turned on by the beauty of these constructs. I also really loved my undergraduate combinatorics course -- especially using generating functions & counting theory.
My bible for learning a lot of math that I never had in courses has been Knuth's 3-volume set. I think that if this set had been available when I was still in high school, I would have gone through and done every exercise on my own.
I could easily see giving up calculus in high school and learning more algebra -- including how to use a computer symbolic algebra system -- and more geometry in the form of computer graphics. I never understood why mathematicians of the middle 20th century wanted to wring all of the pleasure out of math by teaching everything so abstractly; a healthy dose of actual number crunching (aided by a computer, of course) is a much better preparation for calculus. (Perhaps I'm too fond of 19th century mathematics.) Also, matrix algebra -- taught correctly -- can be quite beautiful in its own right, and with excellent computer software available, it should be accessible to high school students.
At 03:59 PM 3/11/2011, rcs@xmission.com wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Where is "Here"? Thanks, Bill C ----- Original Message ----- From: David Makin [mailto:makinmagic@tiscali.co.uk] Sent: Friday, March 11, 2011 10:22 PM To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] High school calculus? Here Calculus has apparently been removed from the National under-16 syllabus which I find a little annoying (especially when successive Governments have claimed there's been no "dumbing down" of education in order to improve pass rates) *because calculus hasn't been replaced by something else*. IMHO the whole of education needs a complete overhaul - it should be designed so the same knowledge is acquired but with 100* the fun factor from the students perspective - to me this means rethink the teaching method completely so it's all done through games and challenges for the students rather than dry lecturing/dictation/textbook use etc. IMHO the real key being a massive increase in *expectations* for all years from 6 to 11. Plus get rid of the incredibly stupid and crippling modern "PC" attitude of removing all sense of competition as if it's morally or ethically incorrect - just congratulate the "winners" and commiserate with the "losers", to do otherwise teachers children something that simply is a lie and some will be unprepared for the naked truth once they leave education. (I realise that this 'PC" practice only applies to some and not all). On 12 Mar 2011, at 00:49, Henry Baker wrote:
Re _not_ teaching calculus in high school:
I have a great deal of sympathy for this view.
Some background: I was a member of our high school's math team that placed highly in the state of Ohio (first, I think, but my memory about this event isn't very good), and I won the Cincinnati math contest my senior year. My senior year, I took all AP courses (except English!), including AP math, which was calculus, more-or-less. As a result of these courses, I was able to "place" out of some courses my freshman year at MIT, including 18.01. My oldest sister's boyfriend was a college math&physics major (who later got his physics doctorate from Heidelberg), and he tried to get me to understand some of the subtleties of "epsilon-delta" calculus, but I have to say that I was not ready for it, and I didn't really understand it.
In retrospect, I should have stopped wasting all of my high school teachers' time and gone straight to MIT after my junior year in high school. Even though my high school had some of the best high school teachers in the U.S., they still were a far cry from what MIT had to offer. When I arrived at MIT & started learning _real_ math & _real_ physics, I felt like I had gone to heaven & I knew that I had just been wasting time in high school.
Calculus never excited me much in high school (or MIT, for that matter). I was far more interested in algebra & mathematical logic. For me, the coolest thing about calculus was the algebraic operations of derivation and antiderivation. I never really "understood" infinitesimal calculus until I took probability theory and especially advanced mathematical logic, where I got to understand models and cardinal numbers and different sizes of infinities. I took an extracurricular number theory course while I was still in high school and _loved_ it. Later, at MIT, I took an algebra course where we studied groups & rings & fields (including finite fields for coding theory), and was really turned on by the beauty of these constructs. I also really loved my undergraduate combinatorics course -- especially using generating functions & counting theory.
My bible for learning a lot of math that I never had in courses has been Knuth's 3-volume set. I think that if this set had been available when I was still in high school, I would have gone through and done every exercise on my own.
I could easily see giving up calculus in high school and learning more algebra -- including how to use a computer symbolic algebra system -- and more geometry in the form of computer graphics. I never understood why mathematicians of the middle 20th century wanted to wring all of the pleasure out of math by teaching everything so abstractly; a healthy dose of actual number crunching (aided by a computer, of course) is a much better preparation for calculus. (Perhaps I'm too fond of 19th century mathematics.) Also, matrix algebra -- taught correctly -- can be quite beautiful in its own right, and with excellent computer software available, it should be accessible to high school students.
At 03:59 PM 3/11/2011, rcs@xmission.com wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Ooops - sorry - UK, more specifically England/Wales :) On 12 Mar 2011, at 19:07, Cordwell, William R wrote:
Where is "Here"?
Thanks, Bill C
----- Original Message ----- From: David Makin [mailto:makinmagic@tiscali.co.uk] Sent: Friday, March 11, 2011 10:22 PM To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] High school calculus?
Here Calculus has apparently been removed from the National under-16 syllabus which I find a little annoying (especially when successive Governments have claimed there's been no "dumbing down" of education in order to improve pass rates) *because calculus hasn't been replaced by something else*.
IMHO the whole of education needs a complete overhaul - it should be designed so the same knowledge is acquired but with 100* the fun factor from the students perspective - to me this means rethink the teaching method completely so it's all done through games and challenges for the students rather than dry lecturing/dictation/textbook use etc. IMHO the real key being a massive increase in *expectations* for all years from 6 to 11. Plus get rid of the incredibly stupid and crippling modern "PC" attitude of removing all sense of competition as if it's morally or ethically incorrect - just congratulate the "winners" and commiserate with the "losers", to do otherwise teachers children something that simply is a lie and some will be unprepared for the naked truth once they leave education. (I realise that this 'PC" practice only applies to some and not all).
On 12 Mar 2011, at 00:49, Henry Baker wrote:
Re _not_ teaching calculus in high school:
I have a great deal of sympathy for this view.
Some background: I was a member of our high school's math team that placed highly in the state of Ohio (first, I think, but my memory about this event isn't very good), and I won the Cincinnati math contest my senior year. My senior year, I took all AP courses (except English!), including AP math, which was calculus, more-or-less. As a result of these courses, I was able to "place" out of some courses my freshman year at MIT, including 18.01. My oldest sister's boyfriend was a college math&physics major (who later got his physics doctorate from Heidelberg), and he tried to get me to understand some of the subtleties of "epsilon-delta" calculus, but I have to say that I was not ready for it, and I didn't really understand it.
In retrospect, I should have stopped wasting all of my high school teachers' time and gone straight to MIT after my junior year in high school. Even though my high school had some of the best high school teachers in the U.S., they still were a far cry from what MIT had to offer. When I arrived at MIT & started learning _real_ math & _real_ physics, I felt like I had gone to heaven & I knew that I had just been wasting time in high school.
Calculus never excited me much in high school (or MIT, for that matter). I was far more interested in algebra & mathematical logic. For me, the coolest thing about calculus was the algebraic operations of derivation and antiderivation. I never really "understood" infinitesimal calculus until I took probability theory and especially advanced mathematical logic, where I got to understand models and cardinal numbers and different sizes of infinities. I took an extracurricular number theory course while I was still in high school and _loved_ it. Later, at MIT, I took an algebra course where we studied groups & rings & fields (including finite fields for coding theory), and was really turned on by the beauty of these constructs. I also really loved my undergraduate combinatorics course -- especially using generating functions & counting theory.
My bible for learning a lot of math that I never had in courses has been Knuth's 3-volume set. I think that if this set had been available when I was still in high school, I would have gone through and done every exercise on my own.
I could easily see giving up calculus in high school and learning more algebra -- including how to use a computer symbolic algebra system -- and more geometry in the form of computer graphics. I never understood why mathematicians of the middle 20th century wanted to wring all of the pleasure out of math by teaching everything so abstractly; a healthy dose of actual number crunching (aided by a computer, of course) is a much better preparation for calculus. (Perhaps I'm too fond of 19th century mathematics.) Also, matrix algebra -- taught correctly -- can be quite beautiful in its own right, and with excellent computer software available, it should be accessible to high school students.
At 03:59 PM 3/11/2011, rcs@xmission.com wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Also just to clarify when I said:
IMHO the real key being a massive increase in *expectations* for all years from 6 to 11.
I meant years as in years old i.e. ages 6 to 11, not years as in school years ! On 12 Mar 2011, at 23:06, David Makin wrote:
Ooops - sorry - UK, more specifically England/Wales :)
On 12 Mar 2011, at 19:07, Cordwell, William R wrote:
Where is "Here"?
Thanks, Bill C
----- Original Message ----- From: David Makin [mailto:makinmagic@tiscali.co.uk] Sent: Friday, March 11, 2011 10:22 PM To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] High school calculus?
Here Calculus has apparently been removed from the National under-16 syllabus which I find a little annoying (especially when successive Governments have claimed there's been no "dumbing down" of education in order to improve pass rates) *because calculus hasn't been replaced by something else*.
IMHO the whole of education needs a complete overhaul - it should be designed so the same knowledge is acquired but with 100* the fun factor from the students perspective - to me this means rethink the teaching method completely so it's all done through games and challenges for the students rather than dry lecturing/dictation/textbook use etc. IMHO the real key being a massive increase in *expectations* for all years from 6 to 11. Plus get rid of the incredibly stupid and crippling modern "PC" attitude of removing all sense of competition as if it's morally or ethically incorrect - just congratulate the "winners" and commiserate with the "losers", to do otherwise teachers children something that simply is a lie and some will be unprepared for the naked truth once they leave education. (I realise that this 'PC" practice only applies to some and not all).
On 12 Mar 2011, at 00:49, Henry Baker wrote:
My son's school does almost exactly what your last paragraph suggests, but it is a private, very progressive school. It offers abstract and linear algebra (in alternate years) to juniors and seniors. I sat in on the current abstract algebra class, and watched as the students played with permutation groups and learned inverses in a way few high school or college students do. Calculus, however, is still the main line for most students. When I asked why this was so, the answer was AP exams, university admissions, and therefore the parents. It seemed so antiquated, the teachers agreed, as has every mathematician I've spoken to since. My background is sort of the compliment of yours. I attended Hawaii public schools from fifth grade onward, except 9th grade. Though I had the highest scores possible on their placement exams, I was put into the lowest math classes from 7th grade on. 9th grade I was in Minnesota where the principal came down to shake my hand for doing so well on the placement exam; the algebra class there showed me how enjoyable math could be. It was real, concrete, and elegant. It made the next two years fighting the system back in Hawaii, where they attempted to put me in a lower math class again, possible. I fell into mathematics because Hawaii taught me to take higher math courses (passing out on a cat during surgery helped push me out of ). Hawaii is an odd place and, at least the schools I attended, had serious issues in math ed. I can't say that these issues were caused by the push towards calculus, but they weren't helped by them. Perhaps more breadth in the curriculum would have added breadth to the mindset of the faculty. Anna On 11 Mar 2011, at 19:49, Henry Baker wrote:
Re _not_ teaching calculus in high school:
I have a great deal of sympathy for this view.
Some background: I was a member of our high school's math team that placed highly in the state of Ohio (first, I think, but my memory about this event isn't very good), and I won the Cincinnati math contest my senior year. My senior year, I took all AP courses (except English!), including AP math, which was calculus, more-or-less. As a result of these courses, I was able to "place" out of some courses my freshman year at MIT, including 18.01. My oldest sister's boyfriend was a college math&physics major (who later got his physics doctorate from Heidelberg), and he tried to get me to understand some of the subtleties of "epsilon-delta" calculus, but I have to say that I was not ready for it, and I didn't really understand it.
In retrospect, I should have stopped wasting all of my high school teachers' time and gone straight to MIT after my junior year in high school. Even though my high school had some of the best high school teachers in the U.S., they still were a far cry from what MIT had to offer. When I arrived at MIT & started learning _real_ math & _real_ physics, I felt like I had gone to heaven & I knew that I had just been wasting time in high school.
Calculus never excited me much in high school (or MIT, for that matter). I was far more interested in algebra & mathematical logic. For me, the coolest thing about calculus was the algebraic operations of derivation and antiderivation. I never really "understood" infinitesimal calculus until I took probability theory and especially advanced mathematical logic, where I got to understand models and cardinal numbers and different sizes of infinities. I took an extracurricular number theory course while I was still in high school and _loved_ it. Later, at MIT, I took an algebra course where we studied groups & rings & fields (including finite fields for coding theory), and was really turned on by the beauty of these constructs. I also really loved my undergraduate combinatorics course -- especially using generating functions & counting theory.
My bible for learning a lot of math that I never had in courses has been Knuth's 3-volume set. I think that if this set had been available when I was still in high school, I would have gone through and done every exercise on my own.
I could easily see giving up calculus in high school and learning more algebra -- including how to use a computer symbolic algebra system -- and more geometry in the form of computer graphics. I never understood why mathematicians of the middle 20th century wanted to wring all of the pleasure out of math by teaching everything so abstractly; a healthy dose of actual number crunching (aided by a computer, of course) is a much better preparation for calculus. (Perhaps I'm too fond of 19th century mathematics.) Also, matrix algebra -- taught correctly -- can be quite beautiful in its own right, and with excellent computer software available, it should be accessible to high school students.
At 03:59 PM 3/11/2011, rcs@xmission.com wrote:
This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing.
Rich
------ From: Anna Johnston [jannaston@gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus
Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are:
(1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening.
(2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA.
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well.
The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum?
Cheers!
Amy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Instead of answering the question directly, I'll talk about this a little, and some of my own high school experiences. I graduated from high school in 2008, so the things I say are current & relevant...at least around here (suburbia of St Paul, MN, USA). There was one time when underlying mathematics was emphasized: intuition, concepts, etc., and that was in maybe 2nd grade of elementary school. One of the most memorable things was this. We had magnets with our names on them, and in the morning, we always put our magnet on the currect side of the side of a metal file cabinet. On the left, we have "Hot Lunch" (school-provided lunch), on the right, "Cold Lunch" (you brought your own lunch). This was so the school could estimate how much to make during the day, I guess. Anyway, one day, there were nine magnets on the "cold lunch" side. The teacher then asked me this question in the morning: "Robert, I have nine magnets for cold lunch. Can I arrange the magnets into a square shape?" Now, we haven't studied anything like this before, so this wasn't a drill question at all. Since I was being "singled out" in a way, I was nervous and didn't know what to say. But the teacher encouraged the class to think about it. We learned a little about multiplication, and I said a bit later "well if it is a square, then the number of lines [rows] should be the same as [the number of] tall lines [columns]" and then "we take those and times them and the number we get is a number we can make squares with". My logic was all mashed up and confusing when I said it, as you can imagine, but we all learned this idea of "square numbers", when knowing only a little bit about what multiplication is, and such. It was a time where we *really* learned, and we didn't memorize anything. Skip to third grade, that's when memory+drill started. Times tables, timed tests, all that stuff. 3: addition, subtraction, multiplication, division, timed tests, long division 4: weights and measures, time, "word problems", area, volume, basic geometry 5: more basic geometry, number properties, calculations/calculators, basic algebra problems 6: more of the same, graphs 7: more of the same, basic algebra, deciphering word problems, "exploratory" stuff 8: Algebra I: solving stuff, quadratics, modeling, graphs At this point, I personally went the "advanced route". Usually you'd only have a year of calculus if you went the basic route. Even then, calculus was optional and you could replace with something else. 9: Algebra 2 + Geometry: more algebra, geometry, two-column proofs 10: Precalculus: trigonometry, circles, triangle solving, sinusoids, vectors, matrices, graph transformations (XY plots, not "edge/vertex" graphs) 11: Calculus 1: limits, differentiation, methods of diff, applications, riemann sums, integration, basic methods, fundamental theorem 12: Calculus 2: methods of integration, related rates, differential equations That's a brief outline of how things went. Now, I want to emphasize *how* it was taught. 3: drill drill drill, you better know 1 digit multiplication and 2-digit-1-digit division 4: This was memorization, but these were practical things you needed to memorize, like grams and kilograms and hectometers. There was terminology like "parallel" and "perpendicular" 5: Memorize area formulas, memorize divisibility, learn how to factorize integers, learn how to do multi-digit multiplication 6: Memorize how to do more problems, memorize how to properly make bar graphs, histograms, box-and-whisker plots, etc. 7: This was not very memorize-oriented. This was more concept oriented. Good teacher. 8: Algebra drill. Factoring, FOIL, expansion, formulas, all that. 9: Memorize how to do two column proofs + the names of logical arguments ("modus ponens", modus tollens (spelling?), contrapositive), learn about reflection, dilation, etc, memorize how angles relate 10: Memorize trig identities, trig values 11: Memorize limit laws, differentiation laws, integration rules 12: Memorize integration rules, differential equation junk Anyway, the point is, above all, everything is so drill based. Everything is so memory based. If you don't remember, you're penalized (exams). This is an obvious problem that's been pointed out a lot, so I won't dive into that issue. There is another issue I had though. I was largely self-taught. I bought my own books and learned all sorts of stuff on my own. The biggest motivating book for me was "Mathematics: From The Birth of Numbers" by Jan Gullberg. This was and is a wonderful compendium of mathematics, written by someone who also taught himself. You can read my review of it here: http://www.amazon.com/review/R2QT66JA1XGJWG Anyway, being self-taught means I could go ahead of the game. There were two huge roadblocks though from letting me get ahead formally. The first one was that to get ahead "formally" (skip grade levels, etc), I had to know what they were teaching. This sounds innocent and fine, but in reality, it is ridiculous. I could demonstrate that I knew how to do trigonometry, but since I couldn't make a box-and-whisker plot proper, that means I did not know enough to move on. But multiply this by 1000. There were lots of these little random things with no importance to math or anything at all, but were required to know -- that most forget a week after anyway. So what if I can't do a compass-straightedge construction of something when I'm in 9th grade. I know the core curriculum, why can't I move on? That was a small one, but it's largely forgotten about and shadowed by an even larger one. I could competently do calculus in 9th grade. I took the AP Calculus BC exam then, and scored a '3'. Not a wonderful score, but for being self-taught entirely, I thought it was grand. A lot of colleges would probably accept that. The better universities, probably not. But I knew what they tested for and I could get a better grade next year anyway. And the stuff I suspect I missed were mundane things anyway. Anyway, so I "basically" have college credit for one semester (one year?) of university calculus (two years of high school), which is single-variable calculus. I also got the top score on the AMC12 (American Mathematics Competition 12) in my school, when I was in 9th grade. I asked the school if, in 9th grade, I could perhaps go to the calculus classes at least, or maybe even go to the university for math classes. They said "no". I argued with them on and on (by 'I', I mean 'we', my father and I), and they granted me to take a test on algebra and geometry. I took it and was pretty sure I did great, but they said I failed it, but refused to show me either the test or the score. A year passed, 10th grade trig. I looked into alternatives, bla bla, asked again if I could skip, nothing came about. In this time, I taught myself some interesting things like Fourier series and began to acquaint myself with some programming stuff. Another year passed, 11th grade calculus. One day in class, I had *my* multivariable calculus and vector algebra book with me. It was just off to the side and the teacher saw it, and he said "wow, why do you have this?" I told him why and told him how I'd already taken the AP tests in 9th grade -- the very tests that the calculus class I was in was preparing us for. He said "well what are you doing here?!" and he suggested I stay after class. I stayed after and he brought me into the other room with the other math teacher, "Mr Butler", who is essentially the head of the high school's math dept. The teacher said "did you know Robert is studying multivariable calculus on his own, and [...]?" The teacher seemed genuinely excited for me; I was pretty happy about it! But then Mr Butler just kind of mumbled saying "yeah ok". My teacher's attitude changed so sharply at this point. My teacher said "we'll look into something Robert, thanks". Anyway, time passed and nothing came of it. I asked about it, and he said he will give me a test. Some time later, I took the test (which covered one year of high school calculus), and he said I passed with a remarkable score. He said I'd have to take another one though. The other teacher, Mr Butler, is the one who administered the second test, which was on "Calculus II" (high school year 2 calculus). I took it and after maybe 2 weeks, I asked about it, and he said I failed, and wasn't able to do anything. It's now one semester through my third year of high school, and nothing has happened. I'm still in Calculus I at this point. After fighting the school a good amount, they finally gave in, and said I could take a university class if I could find one. They didn't help at all with that, so I decided to just take an all-inclusive "Independent & Distance Learning" course at the University of Minnesota, for Multivariable Calculus. After getting that straightened out, the school year is almost over. Maybe 3/4 of the way through. I now go to the library instead of calculus class. I asked the administration if I got credit for calculus 1 and 2, and they said "no." I thought I had "skipped"/got credit for them, but they said no. This caused a big stir. What's the point if skipping if I won't get the necessary *high school* credit to graduate? Toward the end of the year, I finished the entire Calc 3 curriculum in one week. Two days to complete all ten homework submissions (typed up), and I scheduled three consecutive days to complete the two midterms and final exams. I finished that course with 98.9%. During my last year of high school, there seemed to be no opportunities left. There were no math classes to take without the school's help, etc etc. It took the entire year, with a threat of a legal battle (lawyers got involved and whatever), and they gave me credit for calculus 1 and 2, and credit for the four years of French which I completed in one year, and four years of Spanish which I completed in two years. All in all, the whole thing was hell. There was so much time wasted, so much money wasted. It personally affected me. I did poorly in some of my other classes just because of its affect on my mood (I did okay on exams, but in high school, homework accounts for at least 80% of the grade). I barely got through everything. In fact, I had to go to summer school twice because I failed two classes. While it was me who failed some of the classes, I don't carry all of the blame. It was, without a doubt, very stressful and very hard, fighting with the administration during the whole time, trying to get things straightened out and comfortable. In the process, I also moved to a new house, and there were some other family/personal issues to deal with. Applying to the University of Minnesota and some other universities, three or so years ago, was not a success. I was rejected from all of the universities to which I applied, as a result of poor marks in high school, I suppose. My only option right now seems to be (1) go to a community college, and wade through their system, and try to transfer, with only some of the credits transferring to a bigger university, (2) try to figure out a shortcut in the system, and get accepted to graduate school without an undergrad, (3) don't go to college. #1 is the one I dread most. I don't want to go through high school again (which is essentially what a CC is like). And there's no way I could push myself at a CC with advanced courses. I want to learn, not idle. #3 is a very tough choice. I have been applying for jobs for a long time, and have not been accepted to anything. I've done freelance stuff, but it only brings in a very small fraction of what is necessary to live. You really need to strike gold to get off with #3 it seems, these days. #2 seems like a very difficult, but *possible* outcome. I've been auditing graduate math classes at the University of Minnesota for a bit, and my hope is I'll get a connection. Or maybe from somewhere online. I also am trying to get stuff done with my book, and trying to develop publishable material (that isn't "salami science"/smallest unit of publishable material). Anyway, if you've read this far, I applaud you. I kind of went off on tangents. But my entire point is that the high school system around here is tailored to a certain group, and everyone outside of this "box" are essentially outcast. @Thane Plambeck AP exams aren't very good, I don't think. They make teachers teach "to the test". So in the end, things aren't taught, they're drilled. Voluntary CLEP exams are a much better idea I think (which can be taken almost any time), and don't have this whole air around it like AP exams do.
participants (11)
-
Adam P. Goucher -
Anna Johnston -
Cordwell, William R -
David Makin -
Gareth McCaughan -
Gary Antonick -
Henry Baker -
quad -
rcs＠xmission.com -
Robert Munafo -
Thane Plambeck