As mentioned above, I'm revising a set of Frequently Asked Question (FAQ) documents that I prepared several years ago to answer questions from parents, so I thought I'd share my latest drafts here in case they answer any of your questions. I'd love to hear your comments about the clarity, completeness, accuracy, or usefulness of these documents. If you know of other sources to add to the documents, I'd be especially glad to hear about those. This FAQ relates to a frequently discussed issue in gifted education (there is a thread about the issue here on Gifted Issues Forum), and you may be surprised by what I found when I cite-checked the most frequently mentioned source on the issue in the gifted education literature.
2) REPETITION AND PRACTICE
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?
EXECUTIVE SUMMARY: A widely repeated claim is that gifted learners are harmed in their mathematics learning by too much repetition. There is no research to back up this claim. Rather, the best research shows that the best mathematics learners never cease tacking many, many, many challenging problems at all stages of mathematics learning.
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, Ph.D., Director of Research for the Johns Hopkins University Center for Talented Youth (CTY) to check the statements made in the book (which followed up on a suggestion made earlier by the book author about where to find more information on the issue).
Because Carol Mills has received a Ph.D. degree in psychology, I will refer to her as Dr. Mills in the rest of this FAQ. I wrote to her to check the statements in the widely quoted book about gifted education, because the book's author said at a public seminar that the statements were based on findings from Johns Hopkins University Center for Talented Youth research studies. I identified myself as a parent of a CTY student and independent researcher on education issues, especially homeschooling gifted children. I told Dr. Mills that most times when I interact with parents and discuss "practice" of skills that gifted children are learning, I see 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. Whenever anyone cites a source as this issue comes up, the source cited is always the same, namely the book by the author who suggested that I direct follow-up questions to CTY. A frequently cited Web site summarizing the views of the author includes these statements:
* 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 mentioned to Dr. Mills that as I wrote to her I had the author's book at hand, and provided page citations and full, in-context quotations of related statements as they appear in the author's book. I further mentioned that I had gone on two occasions to the largest academic library in the state of Minnesota to check the author's cited references, and the references do not support those statements, at least not as I read them. I discussed each reference, including miscited references, in detail, and noted that an article by Dr. Mills herself
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)
appears in the book's bibliography. The point I emphasized the most in my email to Dr. Mills is that when I check the references I find that they don't back up the conclusions that the author has drawn from them. So I asked Dr. Mills specifically if she and her colleagues at JHU CTY had 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?
I suggested to Dr. Mills that perhaps the author had in mind some kind of distinction like the distinction between "problem" and "exercises," but yet I see many parents specifically avoiding involvement in (for example) mathematical Olympiad competitions for their children because they believe "too much drill" is harmful for their children's mathematical development. That, as I mentioned to Dr. Mills, seems to disagree with the findings in another article by CTY researchers.
Kolitch, E. & Brody, L. (1992). Mathematics Acceleration of Highly Talented Students: An Evaluation. Gifted Child Quarterly, 36(2), 78-86.
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 told Dr. Mills I was 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 often see in response citations to the 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 but one of the author's references seem to lead back to JHU CTY researchers. So I asked Dr. Mills directly: "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."
Dr. 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!"]
After reading the kind replies from Dr. Mills and Dr. Stanley, the way I sum up what the research says is that 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 book published after my correspondence with the CTY researchers, summarizing enormous amounts of recent research on the development of expertise, 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. And as one mathematics teacher wrote a century ago, "Mathematics must be written into the mind, not read into it. 'No head for mathematics' nearly always means 'Will not use a pencil.'" Arthur Latham Baker, Elements of Solid Geometry (1894), page ix.
More recent research confirms that practice beyond the level of performing well in a first performance is helpful for learners.
http://mindshift.kqed.org/2012/02/how-much-practice-is-too-much/ "The perfect execution of a piano sonata or a tennis serve doesn’t mark the end of practice; it signals that the crucial part of the session is just getting underway." This fact has actually been familiar to music teachers for generations. It is still new to many K-12 mathematics teachers that the best time to consolidate learners' improved performance with carefully chosen problems for deliberate practice is just as the learners grasp how to solve such problems successfully.
