Karl Bundy has been gracious enough to allow me to post this here:
Here's a FAQ about an issue I hear about a lot on email lists for
parents of gifted children: is "repetition" in school lessons harmful to
gifted children? I've always thought that the very way the question is
posed (or indeed, the way it is glibly answered) in online discussion is
unhelpful--what kind of repetition are we talking about here? Repetition
of what? What would be the mechanism by which repetition would harm
anyone? Why would that operate any differently for gifted learners from
how it operates for other learners?
In all cases when I ask parents to provide references for their beliefs
about repetition in school lessons, they point to the same author, a
person I have met in person and asked about this issue in public
seminars. I will not name the author in this FAQ, because this is not
about personalities, but I will cast doubt on the author's conclusions,
because I have reason to think that the author's conclusions, as
published, are not warranted by research evidence.
Back in 2004 I looked up the best known book by the author who claims
repetition is harmful for gifted learners and checked all references in
that section of the author's book exhaustively at the libraries of the
University of Minnesota. Then I wrote an email on 7 July 2004 to Carol
Mills, Director of Research for the Johns Hopkins University Center for
Talented Youth (CTY) for reasons that will appear below in the email.
Dear Dr. Mills
I'm writing to check some research results that are reported to
originate from JHU CTY research studies. I am a parent [of a CTY
student] and an independent researcher on education issues, especially
homeschooling gifted children.
In email lists for parents of gifted children I have seen on several
occasions parents suggest that what gifted children most need is to be
advanced as rapidly as possible to the next course in the standard
curriculum rather than to learn each subject in depth through deliberate
practice. Many times when this issue comes up, the source cited is the
same, namely [author]. A frequently cited Web site summarizing the views
of [author] is
[World Wide Web link omitted]
* The learning rate of children above 130 IQ is approximately
8 times faster than for children below 70 IQ
* Gifted students are significantly more likely to retain
science and mathematics content accurately when taught 2-3 times faster
than "normal" class pace.
* Gifted students are significantly more likely to forget or
mislearn science and mathematics content when they must drill and review
it more than 2-3 times
* Gifted students are decontextualists in their processing,
rather than constructivists; therefore it is difficult to reconstruct
"how" they came to an answer.
The third point above, that "Gifted students are significantly more
likely to forget or mislearn science and mathematics content when they
must drill and review it more than 2-3 times," prompted curiosity on my
part about how such a conclusion could be evidenced through research,
and exactly what kind of "drill and review" was in mind.
I have at hand [author]'s book [title and citation]. [Author] has
statements corresponding to the first, second, and third quoted
statements above on pages 281 and 282 in [author's] book, which I will
quote here:
"Pacing and Time Telescoping
"Dr. Brian Start of the University of Melbourne has spent the last
10 years measuring the comparative learning rates of children of varying
abilities and has concluded that the learning rate of children above 130
IQ (gifted) is approximately eight times faster than for children below
70 IQ (mildly mentally impaired) (Start, 1995). Professor Julian Stanley
and his colleagues at Johns Hopkins have suggested that mathematically
precocious students are significantly more likely to retain science and
mathematics content accurately when it has been presented two to three
times faster than the "normal" pace of a traditional mixed-ability class
(Stanley, 1993).
"Furthermore, Stanley has found that gifted students are
significantly more likely to forget or mislearn science and mathematics
content when they are forced to review and drill with it more than two
or three times. In other words, the constant repetition of the regular
classroom, so necessary for mastery among the general population, is
actually detrimental to long-term storage and retrieval of technical
content for gifted students.
"Some preliminary work at the Center for Talented Youth at Johns
Hopkins (Durden, 1992) suggests that this need for less repetition and
drill may also hold true in the areas of foreign language learning,
literature, and writing. The implications seem clear that curriculum
compacting should be an integral part of any educational plan for gifted
students since compacting eliminates unneeded repetition of information
and concepts already mastered, buying time for advanced explorations and
investigations. Research on the academic effects of compacting was
presented in Chapter 5."
I have gone to the University of Minnesota Library on two occasions to
check [author]'s cited references, and the references do not support
those statements, at least not as I read them. The references in the
back of [author]'s book are sparse, and there is an error in citation in
one of the references, and an entire omission of one of the other
references, namely what I think would be properly cited as
Mills, C. J., & Durden, W. G. (1992). Cooperative learning and ability
grouping: An issue of choice. Gifted Child Quarterly, 36 (1), 11-16. (EJ
442 997)
from [author's] bibliography. (You are the same C. J. Mills who wrote
that article, aren't you?) The most important point is that when I check
the references I find that they don't back up the conclusions that
[author] has drawn from them. I heard [author] speak at a public meeting
here in Minnesota on [date] and specifically asked [author] about this
issue of "repetition" in learning for gifted students, and [author] was
unable to provide any more specific reference to the primary research
literature on this point. Rather, what [author] said, when I put
[author] on the spot with a public question, was that any of the many
annual reports from the JHU CTY research group will make this same
point, that repetition is harmful for gifted learners.
What is your sense of the research? Have you and your colleagues at JHU
CTY indeed found that "the constant repetition of the regular classroom,
so necessary for mastery among the general population, is actually
detrimental to long-term storage and retrieval of technical content for
gifted students"? How would such a proposition be demonstrated (that was
my original concern--checking the nature of the research study) if
indeed it has been demonstrated?
My personal research on education of gifted students revolves largely
around mathematics pedagogy in a homeschooling setting, because my
oldest son has for several years had a self-declared passion to learn
pure mathematics. In the writings of mathematicians I have frequently
encountered a distinction between "exercises" and "problems," with the
clear implication that one can have too many exercises, but one can
never have too many problems while learning mathematics. (I can provide
references to some of those writings if you would like, although I
suspect you are more familiar with this literature than I.) Perhaps
[author] has some kind of distinction like this in mind, and yet I see
many parents specifically avoiding involvement in, say, mathematical
Olympiad competitions for their children because they believe "too much
drill" is harmful for their children's mathematical development. I'm not
sure that I can agree with that conclusion based on what I know today,
and especially because I have read the article
Kolitch, E. & Brody, L. (1992). Mathematics Acceleration of Highly
Talented Students: An Evaluation. Gifted Child Quarterly, 36(2), 78-86.
This article appears to be by colleagues of yours at Johns Hopkins.
Noteworthy in the Kolitch & Brody (1992) article is the following
statement about practice in mathematics outside of school classroom
requirements (page 82):
"These students were highly involved in mathematical activities
outside the classroom. Only 2 of the 43 students did not report any
involvement in mathematics competitions. To varying degrees, students
participated in school math teams; state and regional math competitions;
MathCounts; the American High School Mathematics Examination; the USA
Mathematical Olympiad; and other tests, contests, and competitions. . .
. In addition, several students captained math teams, and 3 students
were responsible for organizing teams."
That sounds exactly contrary to the idea that too much practice is
harmful. That sounds like getting a lot of practice is a distinctly good
idea.
So I am puzzled. When I suggest to parents, in online discussion, that
gifted learners, like all learners, get better at what they are learning
if they practice it, I see in response citations to [author]'s
statements, suggesting that practice (taken to be synonymous with the
"repetition" mentioned in her writings) is not helpful for gifted
learners, and indeed harmful for them. And all of [author]'s references
but [author's] reference to Brian Start in Australia seem to lead back
to JHU CTY researchers. [Author] says, when asked a public question
about the basis for her conclusions, that the primary research was done
at and reported by JHU CTY. What are the correct citations, if any, for
research studies that show a harm to gifted learners from "repetition"?
Exactly what was being repeated? How was the success of the learners
under different treatments measured? Would it be fair to characterize
mathematics competitions as NOT "repetitive," because of the great
variety of problems to which they expose young people? I would like to
know what the research you and your colleagues have conducted says about
this issue, because I want to be sure to be as sound as possible in
educating my son, and in advising other parents I meet in person and online.
I appreciate your help with this inquiry in the midst of your extremely
busy schedule. Thanks for all that you do to improve the education of
gifted young people.
[I wrote as above, except that I named the author and title and date I
have edited out of the text of my 7 July 2004 email. Professor Mills
replied to me as below in a 12 July 2004 email.]
Dear [Mr.] Bunday,
As Director of Research for CTY, I will try to respond to your thoughts
and questions regarding the research done at CTY. I have asked Dr.
Julian Stanley and Dr. Linda Brody to also respond to your e-mail
directly. I don't want to speak for them.
Your e-mail raises a number of points, but I will try to respond as
succinctly as possible to what I believe are your major concerns.
From all of our years of working with and studying gifted students, we
know that academically talented students can master content faster than
less able students. And, they can certainly master mathematics content
faster than it is typically taught in the regular classroom. We know
this to be true because we have seen it demonstrated time-after-time in
our summer and distance education classes.
This faster pace of mastering content is, of course, tied to needing
less repetition of the same level of content. Level and pace are the
two major issues here. If children are allowed to learn at a pace that
is somewhat matched to their ability and able to proceed to higher level
content that is more developmentally appropriate for their level of
ability, the pace will begin to slow somewhat and the need for more
practice will increase.
What I think is missing as you interpret [author]'s position and try to
reconcile it with your research and experience is the issue of level and
difficulty of content. Math competitions, particularly Math Olympiad,
involve high-level problems. Practice doing such problems, as you note,
is very beneficial. We would agree with this.
We certainly do not advocate moving gifted children as rapidly as
possible through the standard curriculum and we are certainly not
advocating that they do not study a subject in depth. We believe in
mastery of material before moving on. Depth and breadth of learning are
also both very important, as is some adjustment of pacing and the
ability to move on to higher level content. The appropriate amount of
repetition and practice is whatever moves an individual child to a
mastery level. It varies by child. An appropriate pace also varies by
child.
Do we have any research evidence that proves that repetition is harmful
to gifted children? The short answer is "no." Experience, however,
tells us that unnecessary repetition of content for a child who has
clearly mastered that content can lead to a decrease in motivation to
learn, behavioral problems, and a decrease in interest in the subject.
By extension, too slow of a pace and inappropriate repetition of already
learned material can result in some of the negative effects [author]
notes for some students. But, practice of appropriately challenging
problems for highly able children is most surely beneficial and highly
motivating.
As, I am sure you can appreciate, it is very difficult to conduct
controlled experiments to prove some of these assumptions and observations.
I applaud you for going to the original sources to judge for yourself
what was done, what was claimed, and what was said. I wish more parents
had the background to do the same.
I hope this clarifies the issue somewhat for you. If not, please send me
another message with some specific questions.
[Dr. Mills was true to her word and forwarded my original email to
Julian Stanley, the founder of the Center for Talented Youth, who also
replied by a 12 July 2004 email.]
Dear Mr. (Dr.?) Bunday: Perhaps the best answer to your queries is
contained in my article, "Helping Students Learn Only What They Don't
Already Know," In the professional journal Psychology, Public Policy,
and Law, Vol. 6, No. 1, year 2000, pages 216-222. If you don't have
ready access to this publication, please e-mail me your mailing address
and I'll send you a copy. [I was able to find and photocopy that article
at the University of Minnesota Law School Library shortly after
receiving Professor Stanley's reply.]
My main point is that students should learn a topic or course well
and then move on to the next level, such as second-year algebra, after
MASTERING first-year algebra at the pace appropriate for their
mathematical reasoning ability. Repetition of already WELL-learned
material tends to cause frustration and boredom, and, of course, wasted
time and lost opportunities.
. . . .
As for local, regional, national, and international academic
contests, we strongly recommend them for their challenging and social
value. A math-talented youth would usually be well advised to begin
with the elementary school math "Olympiad," if available, and proceed on
in seventh AND eighth grade with MathCounts, followed all the way
through each of the four years of high school with the American High
School Mathematics Examination, leading, IF he or she excels, to the
next levels: invitational contest, USAMO, IMO training camp, and to a
place on the six--person team competing for the United States in the
International Mathematical Olympiad (IMO). Half of the IMO contestants
will win a medal (bronze, silver, or gold). A very few will get special
commendations on one or more problems. A VERY few will earn a perfect
score. There's PLENTY of "ceiling" in this progression. Needless
repetition? Of course not!
. . . .
We strongly advocate regular, systematic achievement testing,
especially via the College Board SAT II series and the 34 excellent
tests of the College Board's Advanced Placement Program . These we
consider CRUCIAL for home-schooled youth.
We try strongly to discourage moving ahead fast in grade placement
and entering college very young, such as less than 16 years old.
Skipping one grade at an optimal place in the progression may be
appropriate. Our experience with the very brightest of our millions of
examinees indicates that multiple grade skipping is unnecessary and
undesirable. We do not object to college courses taken on a part-time
basis while still in high school. Working on one's own, with a suitable
mentor, can make a wide range of AP courses available.
[Professor Stanley was born in 1918 and died a few months after he and I
exchanged a second set of emails. His later advocacy of NOT going to
college at unusually young ages, but rather taking college-level work as
a high-school student, reflected his first generation of experience with
Talent Search students, only a few of whom thrived well after very early
college entrance. I especially appreciate his comment about the
progression of difficulty level in mathematics competitions: "There's
PLENTY of 'ceiling' in this progression. Needless repetition? Of course
not!"]
To sum up what the research says, if there is any harm at all in school
"repetition," it is primarily the harm of
a) missed opportunities to do something harder and more educational
(which, I acknowledge, are opportunities hard to develop in some school
systems)
or
b) the student losing interest and thereafter doing too little practice
to continue advancing in ability. Until mastery is achieved, practice is
wholly beneficial. As mastery of one level of a subject is achieved,
move on to the next level, but keep right on practicing.
A new book, summarizing enormous amounts of recent research on the
development of expertise, which puts things in perspective is
The Cambridge Handbook of Expertise and Expert Performance edited by K.
Anders Ericsson et al.
http://www.amazon.com/Cambridge-Handbook-Expertise-Expert-Performance/dp/0521600812/The "ten-year rule" applies to all learners of all subjects: the only
way to become an expert is to devote ten years (in round figures) of
intensive deliberate practice to mastering the skills and
domain-specific knowledge of a particular domain.
--
- Show quoted text -
Karl M. Bunday, President
Edina Center for Academic Excellence
http://ecae.netkmbunday AT earthlink DOT net (preferred email address)
Here's a FAQ about an issue I hear about a lot on email lists for parents of gifted children: is "repetition" in school lessons harmful to gifted children? I've always thought that the very way the question is posed (or indeed, the way it is glibly answered) in online discussion is unhelpful--what kind of repetition are we talking about here? Repetition of what? What would be the mechanism by which repetition would harm anyone? Why would that operate any differently for gifted learners from how it operates for other learners?
In all cases when I ask parents to provide references for their beliefs about repetition in school lessons, they point to the same author, a person I have met in person and asked about this issue in public seminars. I will not name the author in this FAQ, because this is not about personalities, but I will cast doubt on the author's conclusions, because I have reason to think that the author's conclusions, as published, are not warranted by research evidence.
Back in 2004 I looked up the best known book by the author who claims repetition is harmful for gifted learners and checked all references in that section of the author's book exhaustively at the libraries of the University of Minnesota. Then I wrote an email on 7 July 2004 to Carol Mills, Director of Research for the Johns Hopkins University Center for Talented Youth (CTY) for reasons that will appear below in the email.
Dear Dr. Mills
I'm writing to check some research results that are reported to originate from JHU CTY research studies. I am a parent [of a CTY student] and an independent researcher on education issues, especially homeschooling gifted children.
In email lists for parents of gifted children I have seen on several occasions parents suggest that what gifted children most need is to be advanced as rapidly as possible to the next course in the standard curriculum rather than to learn each subject in depth through deliberate practice. Many times when this issue comes up, the source cited is the same, namely [author]. A frequently cited Web site summarizing the views of [author] is
[World Wide Web link omitted]
* The learning rate of children above 130 IQ is approximately 8 times faster than for children below 70 IQ
* Gifted students are significantly more likely to retain science and mathematics content accurately when taught 2-3 times faster than "normal" class pace.
* Gifted students are significantly more likely to forget or mislearn science and mathematics content when they must drill and review it more than 2-3 times
* Gifted students are decontextualists in their processing, rather than constructivists; therefore it is difficult to reconstruct "how" they came to an answer.
The third point above, that "Gifted students are significantly more likely to forget or mislearn science and mathematics content when they must drill and review it more than 2-3 times," prompted curiosity on my part about how such a conclusion could be evidenced through research, and exactly what kind of "drill and review" was in mind.
I have at hand [author]'s book [title and citation]. [Author] has statements corresponding to the first, second, and third quoted statements above on pages 281 and 282 in [author's] book, which I will quote here:
"Pacing and Time Telescoping
"Dr. Brian Start of the University of Melbourne has spent the last 10 years measuring the comparative learning rates of children of varying abilities and has concluded that the learning rate of children above 130 IQ (gifted) is approximately eight times faster than for children below 70 IQ (mildly mentally impaired) (Start, 1995). Professor Julian Stanley and his colleagues at Johns Hopkins have suggested that mathematically precocious students are significantly more likely to retain science and mathematics content accurately when it has been presented two to three times faster than the "normal" pace of a traditional mixed-ability class (Stanley, 1993).
"Furthermore, Stanley has found that gifted students are significantly more likely to forget or mislearn science and mathematics content when they are forced to review and drill with it more than two or three times. In other words, the constant repetition of the regular classroom, so necessary for mastery among the general population, is actually detrimental to long-term storage and retrieval of technical content for gifted students.
"Some preliminary work at the Center for Talented Youth at Johns Hopkins (Durden, 1992) suggests that this need for less repetition and drill may also hold true in the areas of foreign language learning, literature, and writing. The implications seem clear that curriculum compacting should be an integral part of any educational plan for gifted students since compacting eliminates unneeded repetition of information and concepts already mastered, buying time for advanced explorations and investigations. Research on the academic effects of compacting was presented in Chapter 5."
I have gone to the University of Minnesota Library on two occasions to check [author]'s cited references, and the references do not support those statements, at least not as I read them. The references in the back of [author]'s book are sparse, and there is an error in citation in one of the references, and an entire omission of one of the other references, namely what I think would be properly cited as
Mills, C. J., & Durden, W. G. (1992). Cooperative learning and ability grouping: An issue of choice. Gifted Child Quarterly, 36 (1), 11-16. (EJ 442 997)
from [author's] bibliography. (You are the same C. J. Mills who wrote that article, aren't you?) The most important point is that when I check the references I find that they don't back up the conclusions that [author] has drawn from them. I heard [author] speak at a public meeting here in Minnesota on [date] and specifically asked [author] about this issue of "repetition" in learning for gifted students, and [author] was unable to provide any more specific reference to the primary research literature on this point. Rather, what [author] said, when I put [author] on the spot with a public question, was that any of the many annual reports from the JHU CTY research group will make this same point, that repetition is harmful for gifted learners.
What is your sense of the research? Have you and your colleagues at JHU CTY indeed found that "the constant repetition of the regular classroom, so necessary for mastery among the general population, is actually detrimental to long-term storage and retrieval of technical content for gifted students"? How would such a proposition be demonstrated (that was my original concern--checking the nature of the research study) if indeed it has been demonstrated?
My personal research on education of gifted students revolves largely around mathematics pedagogy in a homeschooling setting, because my oldest son has for several years had a self-declared passion to learn pure mathematics. In the writings of mathematicians I have frequently encountered a distinction between "exercises" and "problems," with the clear implication that one can have too many exercises, but one can never have too many problems while learning mathematics. (I can provide references to some of those writings if you would like, although I suspect you are more familiar with this literature than I.) Perhaps [author] has some kind of distinction like this in mind, and yet I see many parents specifically avoiding involvement in, say, mathematical Olympiad competitions for their children because they believe "too much drill" is harmful for their children's mathematical development. I'm not sure that I can agree with that conclusion based on what I know today, and especially because I have read the article
Kolitch, E. & Brody, L. (1992). Mathematics Acceleration of Highly Talented Students: An Evaluation. Gifted Child Quarterly, 36(2), 78-86.
This article appears to be by colleagues of yours at Johns Hopkins. Noteworthy in the Kolitch & Brody (1992) article is the following statement about practice in mathematics outside of school classroom requirements (page 82):
"These students were highly involved in mathematical activities outside the classroom. Only 2 of the 43 students did not report any involvement in mathematics competitions. To varying degrees, students participated in school math teams; state and regional math competitions; MathCounts; the American High School Mathematics Examination; the USA Mathematical Olympiad; and other tests, contests, and competitions. . . . In addition, several students captained math teams, and 3 students were responsible for organizing teams."
That sounds exactly contrary to the idea that too much practice is harmful. That sounds like getting a lot of practice is a distinctly good idea.
So I am puzzled. When I suggest to parents, in online discussion, that gifted learners, like all learners, get better at what they are learning if they practice it, I see in response citations to [author]'s statements, suggesting that practice (taken to be synonymous with the "repetition" mentioned in her writings) is not helpful for gifted learners, and indeed harmful for them. And all of [author]'s references but [author's] reference to Brian Start in Australia seem to lead back to JHU CTY researchers. [Author] says, when asked a public question about the basis for her conclusions, that the primary research was done at and reported by JHU CTY. What are the correct citations, if any, for research studies that show a harm to gifted learners from "repetition"? Exactly what was being repeated? How was the success of the learners under different treatments measured? Would it be fair to characterize mathematics competitions as NOT "repetitive," because of the great variety of problems to which they expose young people? I would like to know what the research you and your colleagues have conducted says about this issue, because I want to be sure to be as sound as possible in educating my son, and in advising other parents I meet in person and online.
I appreciate your help with this inquiry in the midst of your extremely busy schedule. Thanks for all that you do to improve the education of gifted young people.
[I wrote as above, except that I named the author and title and date I have edited out of the text of my 7 July 2004 email. Professor Mills replied to me as below in a 12 July 2004 email.]
Dear [Mr.] Bunday,
As Director of Research for CTY, I will try to respond to your thoughts and questions regarding the research done at CTY. I have asked Dr. Julian Stanley and Dr. Linda Brody to also respond to your e-mail directly. I don't want to speak for them.
Your e-mail raises a number of points, but I will try to respond as succinctly as possible to what I believe are your major concerns.
From all of our years of working with and studying gifted students, we know that academically talented students can master content faster than less able students. And, they can certainly master mathematics content faster than it is typically taught in the regular classroom. We know this to be true because we have seen it demonstrated time-after-time in our summer and distance education classes.
This faster pace of mastering content is, of course, tied to needing less repetition of the same level of content. Level and pace are the two major issues here. If children are allowed to learn at a pace that is somewhat matched to their ability and able to proceed to higher level content that is more developmentally appropriate for their level of ability, the pace will begin to slow somewhat and the need for more practice will increase.
What I think is missing as you interpret [author]'s position and try to reconcile it with your research and experience is the issue of level and difficulty of content. Math competitions, particularly Math Olympiad, involve high-level problems. Practice doing such problems, as you note, is very beneficial. We would agree with this.
We certainly do not advocate moving gifted children as rapidly as possible through the standard curriculum and we are certainly not advocating that they do not study a subject in depth. We believe in mastery of material before moving on. Depth and breadth of learning are also both very important, as is some adjustment of pacing and the ability to move on to higher level content. The appropriate amount of repetition and practice is whatever moves an individual child to a mastery level. It varies by child. An appropriate pace also varies by child.
Do we have any research evidence that proves that repetition is harmful to gifted children? The short answer is "no." Experience, however, tells us that unnecessary repetition of content for a child who has clearly mastered that content can lead to a decrease in motivation to learn, behavioral problems, and a decrease in interest in the subject.
By extension, too slow of a pace and inappropriate repetition of already learned material can result in some of the negative effects [author] notes for some students. But, practice of appropriately challenging problems for highly able children is most surely beneficial and highly motivating.
As, I am sure you can appreciate, it is very difficult to conduct controlled experiments to prove some of these assumptions and observations.
I applaud you for going to the original sources to judge for yourself what was done, what was claimed, and what was said. I wish more parents had the background to do the same.
I hope this clarifies the issue somewhat for you. If not, please send me another message with some specific questions.
[Dr. Mills was true to her word and forwarded my original email to Julian Stanley, the founder of the Center for Talented Youth, who also replied by a 12 July 2004 email.]
Dear Mr. (Dr.?) Bunday: Perhaps the best answer to your queries is contained in my article, "Helping Students Learn Only What They Don't Already Know," In the professional journal Psychology, Public Policy, and Law, Vol. 6, No. 1, year 2000, pages 216-222. If you don't have ready access to this publication, please e-mail me your mailing address and I'll send you a copy. [I was able to find and photocopy that article at the University of Minnesota Law School Library shortly after receiving Professor Stanley's reply.]
My main point is that students should learn a topic or course well and then move on to the next level, such as second-year algebra, after MASTERING first-year algebra at the pace appropriate for their mathematical reasoning ability. Repetition of already WELL-learned material tends to cause frustration and boredom, and, of course, wasted time and lost opportunities.
. . . .
As for local, regional, national, and international academic contests, we strongly recommend them for their challenging and social value. A math-talented youth would usually be well advised to begin with the elementary school math "Olympiad," if available, and proceed on in seventh AND eighth grade with MathCounts, followed all the way through each of the four years of high school with the American High School Mathematics Examination, leading, IF he or she excels, to the next levels: invitational contest, USAMO, IMO training camp, and to a place on the six--person team competing for the United States in the International Mathematical Olympiad (IMO). Half of the IMO contestants will win a medal (bronze, silver, or gold). A very few will get special commendations on one or more problems. A VERY few will earn a perfect score. There's PLENTY of "ceiling" in this progression. Needless repetition? Of course not!
. . . .
We strongly advocate regular, systematic achievement testing, especially via the College Board SAT II series and the 34 excellent tests of the College Board's Advanced Placement Program . These we consider CRUCIAL for home-schooled youth.
We try strongly to discourage moving ahead fast in grade placement and entering college very young, such as less than 16 years old. Skipping one grade at an optimal place in the progression may be appropriate. Our experience with the very brightest of our millions of examinees indicates that multiple grade skipping is unnecessary and undesirable. We do not object to college courses taken on a part-time basis while still in high school. Working on one's own, with a suitable mentor, can make a wide range of AP courses available.
[Professor Stanley was born in 1918 and died a few months after he and I exchanged a second set of emails. His later advocacy of NOT going to college at unusually young ages, but rather taking college-level work as a high-school student, reflected his first generation of experience with Talent Search students, only a few of whom thrived well after very early college entrance. I especially appreciate his comment about the progression of difficulty level in mathematics competitions: "There's PLENTY of 'ceiling' in this progression. Needless repetition? Of course not!"]
To sum up what the research says, if there is any harm at all in school "repetition," it is primarily the harm of
a) missed opportunities to do something harder and more educational (which, I acknowledge, are opportunities hard to develop in some school systems)
or
b) the student losing interest and thereafter doing too little practice to continue advancing in ability. Until mastery is achieved, practice is wholly beneficial. As mastery of one level of a subject is achieved, move on to the next level, but keep right on practicing.
A new book, summarizing enormous amounts of recent research on the development of expertise, which puts things in perspective is
The Cambridge Handbook of Expertise and Expert Performance edited by K. Anders Ericsson et al.
http://www.amazon.com/Cambridge-Handbook-Expertise-Expert-Performance/dp/0521600812/The "ten-year rule" applies to all learners of all subjects: the only way to become an expert is to devote ten years (in round figures) of intensive deliberate practice to mastering the skills and domain-specific knowledge of a particular domain.
)