Gifted Issues Discussion homepage Forums GT Research Mathematics Education FAQs

Hi, everyone,

I was reminded to visit this forum, which I joined years ago but then neglected, by a Google search that led me to a post here. The search was part of an effort I'm making today to update some FAQ pages on mathematics education. As I'm revising a set of Frequently Asked Question (FAQ) documents that I prepared several years ago to answer questions from parents, 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 clarify, 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. I'll post the first revised FAQ draft in this message, and then post others as replies to this message.



1) PROBLEMS VERSUS EXERCISES

I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount).

Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves.

"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy.

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."

Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.

I have since read about this distinction in several other books.

"Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics:

Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . .

Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . .

Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . .

Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . .

Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems."

Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73.

This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)"


Other books make this distinction too.

"First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . .

"A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it."

Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4.


". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.'

"The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises."

Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii.


"It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance."

Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.


"Learn by Solving Problems

"We believe that the best way to learn mathematics is by solving problems--lots and lots of problems. In fact, we believe the best way to learn mathematics is to try to solve problems that you don't know how to do. When you discover something on your own, you'll understand it much better than if someone just tells it to you.

. . . .

"If you find the problems are too easy, this means you should try harder problems. Nobody learns very much by solving problems that are too easy for them."

Rusczyk, Richard, Patrick, David, and Boppana, Ravi (2011). Prelgebra. Alpine, CA: AoPS Incorporated, p. iii.

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.

Have you read anything by Dr. Linda Silverman and visual spatial learning?

I ask because I have a kindergarten, vsl, eg/pg DS6. I've found that repetition and practice with math is detrimental and causes problems. He's a vsl and he doesn't tend to function well under timed tests or rote with math. Dr. Linda Silverman's research has supported this argument.

Last year, my son was at a gifted school where he was able to accelerate in the math workbook at his own pace. He quickly completed the pre-k/k/1st grade curriculum in math within 2 1/2 months. He learned how to multiply from a visual illustration and with no instruction or practice. This year, he's at a different gifted school and they use https://www.xtramath.org/ and he's refusing to go beyond adding. Part of it is motivation and math being boring and no longer fun, but I think there's possibly more to the story. Next year, I plan to homeschool him because he used to love math.

I find your question and finding interesting. I find my son can pick up things by observing (visual!) and without practice in some cases. He doesn't always learn in a linear or sequential fashion either (another vsl trait!) and yet math instruction is aimed in this manner.

Part of problem seems to be that some eg/pg people tend to rely more heavily on either the right or left hemisphere of the brain and this affects how they learn, worldview, etc. - yet much of the gifted literature (from what I've read) doesn't seem to address how this can impact with learning, worldview, etc. I also wonder if some kids treat it as 'playing' with video games or equivalent and end up becoming enraptured with math.

cdfox, I have to respectfully disagree with the idea that practice (at the appropriate level, as Karl has already pointed out is important) will harm or fail to help your son progress in math. It's likely that one big reason your son needed no practice on pre-K to 1st grade concepts is that he already had a good grasp of them; it's unlikely, though I suppose not impossible, that he would be able to learn math through a postgraduate level instantly at a glance. I would agree that lack of motivation and boredom can be major problems, and I think that's all you've seen. A person cannot unlearn concepts through extra practice; they can only become demotivated.

The way I see it, there is a point of diminishing returns with extra math practice. Benefits of math work in general include learning or refining new math concepts, and learning or refining skills such as computation algorithms or techniques. Learning a new concept as an overview does not necessarily mean that the concept is mastered completely. Being able to demonstrate knowledge of a computation skill doesn't mean that it's mastered either; practice can improve accuracy, speed, etc. Limitations on benefits of math work would include decreasing improvement in accuracy and speed, decreasing improvement in conceptual knowledge by further work at the same conceptual level, and decreasing student involvement due to boredom.

It's obviously true that children with a math bent can learn math concepts more quickly than normal. All I'm saying is that this doesn't mean that they fall off in ability or understanding with more work in general-- it just has to be appropriate work.

I also recall finding some of Linda Silverman's work questionable, for instance her widely repeated assertions about parents being excellent identifiers of giftedness. I'd have to read up on her work with visual spatial learners to form an opinion of what she's said about math practice. I'd be grateful for a link to the research you mentioned, or else I can try to request it through the interlibrary interspouse loan system.

Thanks, this is somewhat helpful and see what you're saying. I'm still not sure if he had a good grasp on 1st grade concepts and needed no practice since he wasn't adding or subtracting before entering the gifted school.

I completely agree with you on the lack of motivation, boredom, diminishing return with extra math, and mastering with computation. Yes. I completely agree with math not falling off in ability or understanding and need more appropriate work. Dr. Silverman makes the point that vsl learners see in patterns more and in terms of relationships so it would seem that once you've got the pattern or relationship mastered there's no fun or something new to learn.

We're on the waitlist to see Dr. Lovecky in the spring and math is one of the topics I have questions on. It's been a bugbear of mine.

http://www.gifteddevelopment.com/ - is Dr. Silverman's website and she's got links on it

Dr. Linda Silverman, Upside-Down Brilliance: The Visual-Spatial Learner.

I think parents can be excellent identifiers if they know their family's history and if a family had scored eg/pg on IQ tests but this kind of information isn't always circulated within families and parents can be blind to brilliance.

Thanks for your reply. I have read several writings by Linda Silverman about visual-spatial learners, as she calls them. I don't think she is the best authority on that subject, and I don't recommend following her advice, but I will give her credit for raising the issue that different learners may have different customary approaches to some subjects as they begin the learning process.

I have attended various seminars and read various articles and books about "learning styles," and every time I am tested on mine, I fit into the "all of the above" category. I definitely have a bigger visual element in my thinking about mathematics and other subjects than many people I know. When I ask my wife (a piano performance major who works as a music teacher) about her learning styles or modes of thinking, she also says "all of the above." Whether in modern foreign languages (my undergraduate major), computer science (my oldest son's major), or music (my wife's major), the really thorough experts learn multiple representations of the subject. My wife learned some very helpful visual representations of music theory through one of her professors

https://www.areditions.com/books/rs002.html

that she teaches to most of her students. The crucial issue about learning styles is that learning styles themselves are learnable, and good education is all about broadening a learner's toolkit so that the learner can apply more approaches to learning new material in whatever domain the learner is learning.



Thanks for your reply too. That's my sense of the evidence, that motivation matters a lot in learning, especially learning difficult subjects, and too many American children grow up fearing hard work in learning. When I was learning Chinese as an undergraduate, I prepared by visiting the Chinese class at my undergraduate university while I was still finishing up high school. I asked the teaching assistant who was leading the first-year class that day if it was required to listen to tapes of Chinese in the language lab. She said, "We don't require attendance in the language lab. But you can always tell who listens to the tapes." That was all she needed to say to motivate me to listen to every inch of Chinese-language audiotapes in the language lab while I was an undergraduate student. That had the gratifying effect of impressing people who grew up speaking the language when I lived overseas (of whom the crucial person to impress was my girlfriend-then-wife) and gaining me professional employment as a Chinese-English interpreter after I returned to the United States from living overseas. I had been lucky that my first year textbook included a preface that said "language learning is overlearning," a saying of linguist Leonard Bloomfield that many brilliant language teachers have taken to heart.

So to help a child keep motivation when they get to the "I know that" level, before they reach the error-free, second-nature level of performance, it is important to let the child know that all the great performers in any domain--from Mozart right on to the present day--have devoted time, effort, and motivation to overlearning their domain. That's how people get to be really good at what they do.

Thank you for this interesting discussion. It is my sense that repetition at the "just right" or even "slightly too easy" level can indeed be very beneficial. My DD "got" borrowing across zeros immediately, but she has benefited from practice, which has given her great automaticity. However, when she is asked to do review math on the single-digit level she rebels. I don't think this is HARMING her mastery of those facts, but it isn't exciting her about math, that's for sure.

For us it's all about the impression. Somethings take a little longer.

Think about it this way. The three little numbers on the back of your credit card. When you first get your card, they are there, easy to read. But a month or two down the road, the ink has worn off. You continue to read those numbers by looking at the impression in the card.

When a child is actively working on a new skill the information is right there. But that initial learning gets pushed aside to move on to other things. There needs to be enough repetition so that an "impression" has been made in the child's memory. So that when the time comes they can recall what they need to move on to higher maths.

For my son this impression comes between 15-20 problems. Anything more than that is overkill, less makes recall iffy. Since everyone learns at different rates, you need to identify what your child's impression line is.

Thanks for the further replies. Here's another FAQ section:


3) LINKS ABOUT LEARNING MATHEMATICS AND OTHER SUBJECTS

I was first introduced to a mathematician writing about how to teach elementary mathematics when a parent told me back in the twentieth century about the article "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education,"

http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf

by Professor Hung-hsi Wu. His writings have been very influential on my thinking about mathematics education. In June 2010, I had the privilege of meeting Professor Wu in person at a teacher training workshop in St. Paul, Minnesota.

EXECUTIVE SUMMARY: The simple things in mathematics are the hard things. Any learner of mathematics, and more generally any learner of any subject, has to know the foundational principles of the subject thoroughly, and there is always more to learn about how the "basics" fit together to make ideas.

A link that furthered my process of pondering how to teach mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma.

http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

Another review of that excellent book by mathematician Roger Howe

http://www.ams.org/notices/199908/rev-howe.pdf

is also food for thought. In some countries, elementary mathematics is not considered "easy" mathematics, but rather fundamental mathematics, which must be understood in full context to build a foundation for later mathematical study.

Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

Richard Rusczyk, a champion mathematics competitor in high school and now a publisher of mathematics textbooks, among other ventures, has written an interesting article "The Calculus Trap":

http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap&

I particularly like this article's statement,

"If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. Students of like interest and ability feed off of each other. They learn from each other; they challenge and inspire each other."

which is one reason to encourage able mathematics learners to learn together. I had the privilege of meeting Richard Rusczyk twice in the summer of 2010, once at a Summit of Davidson Young Scholars program participants, and then again at the Minnesota State High School Mathematics League coaches conference. Rusczyk thinks it is crucial for bright students to avoid the "tyranny of 100 percent," in which they only get school homework assignments that are easy enough to do perfectly. He thinks it is very important for the development of young learners to face problems that are hard enough to challenge a learner, so the learner learns how to persist in problem-solving and not give up too soon.

Another good article about a broader rather than narrower mathematics education is "Mathematics Education." Notices of the American Mathematical Society 37:7 (September, 1990) 844-850.

http://arxiv.org/PS_cache/math/pdf/0503/0503081v1.pdf

by William Thurston, a Fields medalist.

"Another problem is that precocious students get the idea that the reward is in being 'ahead' of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work."

Thurston explains why a broad mathematics education is useful in helping mathematics research advance in his article "On Proof and Progress in Mathematics." Bulletin of the American Mathematical Society, 30 (1994) 161-177.

http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf

Timothy Gowers, a mathematician who is both an International Mathematical Olympiad gold medalist and a Fields Medal winner, wrote "The Two Cultures of Mathematics"

http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

to point out that both solving problems and building and understanding theories are important aspects of mathematics. The development of mathematics is limited when mathematicians only do one or the other.

Terence Tao, answering the question "Does one have to be a genius to do maths?"

http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

says, "The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities."

4) THE EXPLORER IS THE PERSON WHO IS LOST, OR COURAGE IN THE FACE OF STUPIDITY

EXECUTIVE SUMMARY: Learning something really challenging is scary, but everyone has felt that fear, and you can overcome the fear of the unknown to learn more than you first thought you could learn.


In the summer of 2005, my oldest son had the privilege of attending the MathPath program for middle-school-age mathematics learners. Some other parents and I who brought children there were in the audience with the program students for the program's opening lecture. In the Colorado College auditorium where the lecture took place, there was a large blackboard on the speaker's stage. Mathematician Paul Zeitz stepped on to the stage, and wrote an enormous word on the blackboard:




STUPID




Everyone in the room wondered what kind of lecture would follow the lecturer putting that word on the blackboard. Zeitz began his lecture by asking members of the audience to think about how long they had ever worked on a mathematics problem for school. Most of the young people in the room seemed to indicate that they had spent at most an hour or so working on a mathematics problem for school homework. As the lecture continued, the idea came out that most school mathematics problems can be solved in just a few minutes by a bright learner. Maybe an especially tough mathematics problem from school might take as long as an hour. But Zeitz pointed out that professional mathematicians work on problems that can take days, weeks, months, or even years. Some famous mathematical problems have remained unsolved for centuries, with mathematicians working on a problem that was first brought up before they were born and dying before anyone in the world found out the solution. Zeitz said that if a problem is really a problem, and not just an exercise, the first issue is figuring out where to begin. If a problem is sufficiently hard, and many problems confronted by professional mathematicians are that hard, it may take a long time even to figure out how to start. Zeitz said that the way anyone feels while not knowing how to begin is stupid. Feeling stupid can be very discouraging. Especially if a student is used to getting the answer quickly from school mathematics courses, first working on a professional mathematics problem can be very discouraging and seem impossible.

The most important thing for someone working on a hard mathematics problem is not to be afraid while feeling stupid. When you feel stupid, don't say to yourself, "I'm stupid," but rather "This problem is hard. I'll have to work hard on this, try out lots of different ways to figure this out, and not give up too soon." Just because you feel stupid doesn't mean you are stupid. Sometimes you have to be brave and endure feeling stupid for days, weeks, months, or even years, but if you keep working on the problem, you can break through and feel glad that you didn't give up. Feeling stupid is a sign you are working on a hard problem. It's not a reason to give up.

When I showed a first draft of this essay to my wife, a piano teacher, she commented that much the same kind of courage applies to studying music. After thinking about that for a while, I realized that the self-talk that comes to mind for music students who feel stupid when they think a piece is too hard is "I have no musical talent." But I have seen several examples among various performance students who, at first, seemed stuck on a piece for weeks at a time, practicing over and over and over and never sounding musical, and then having a "breakthrough" so that they sounded completely different, much more musical, every time afterward when they played. Musical "talent" can develop after learning plateaus throughout the course of a lifetime of studying music.

I then recalled that another author related the idea of a specific lack of talent, rather than more general "stupidity," to learning plateaus in the study of mathematics. Many gainfully employed adults think they don’t have a brain for mathematics. Mathematician W. W. Sawyer wrote in his book Vision in Elementary Mathematics (original publication by Penguin 1964, reprint by Dover 2003) about parents who spoke to their children about not having talent in mathematics. Sawyer firmly believed, based on his experience as a mathematics teacher in various countries, that anyone can learn more about mathematics if properly taught. So Sawyer wrote, "The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" [Sawyer 1964:5] In other words, don't set up your children for believing that they cannot succeed in mathematics until you have given them a chance to learn from a good teacher. An even more famous mathematician than Sawyer who wrote interesting correspondence study books about secondary school mathematics, first in Russian and then in English, was Israel M. Gelfand (1913-2009). Gelfand wrote, "Students have no shortcomings, they have only peculiarities. The job of a teacher is to turn these peculiarities into advantages." So if you are young, and feel stupid because you don’t know how to begin in learning a hard subject, first of all look for a teacher who believes you can learn the subject, and who can help you make connections between what you already know and what you need to learn.

A new research publication, "The Neurodevelopmental Basis of Math Anxiety,"

http://stanford.edu/group/scsnl/cgi...Neurodevelopmental_Basis_Math_Anxiety_12

suggests, based on data gathered from brain scans in children, that mathematics anxiety is a learned fear much like other specific phobias. From this point of view, the most important thing a teacher can do for a young mathematics learner is dispel fear, encourage boldness, and focus the learner’s attention on problem-solving skills rather than the negative emotions that derive from fear.

I recently checked my memory of Paul Zeitz's lecture based on the word "STUPID" by emailing Professor Zeitz. He was very glad to hear that some of my mathematics pupils have learned to take "stupid" as an inspirational word, writing it on the whiteboard to cheer themselves on before they play mathematical games against me or against one another. They have lost the fear of taking on tough challenges, and now delight in working on problems that are hard.

Professor Zeitz told me that he now sums up the advice he gave in his 2005 lecture with a phrase he takes from Tim Cahill, founding editor of Outside magazine: "The explorer is the person who is lost." Zeitz used that line in the foreword he wrote for the 2011 Math Survey of Stuyvesant High School in New York City, his alma mater, in which he wrote,

"In 1998, when I wrote the first edition of The Art and Craft of Problem Solving (whose introduction specifically mentions Stuyvesant!), I included an epigraph that was a quote from Jaguars Ate My Flesh, a collection of humorous travel essays by Tim Cahill. The quote was:

"The explorer is the person who is lost.

"The meaning, as we mathematicians like to say, is 'clear.' You cannot accomplish anything meaningful without making mistakes. In fact, even if you don’t accomplish much that is meaningful, you need to make mistakes. This is a sometimes counterintuitive idea for many Stuyvesant students, trained and rewarded over many years not to make mistakes.

"One of the hardest things to do, especially for us bright high achievers, is to get used to being not just stupid, but bumblingly stupid. All mathematicians spend most—usually way above 90%—of their waking hours feeling confused at best. That is not a bad feeling, mind you, although it is less good than the rare moments of insight. After all, mathematicians are explorers, and explorers are lost. It’s an existential condition of being an explorer.

"So, learn to enjoy the sensation of being lost. As you encounter mathematical problems—and here I distinguish problems, questions that you do not, at the outset, know how to approach, from exercises, which are the things at the end of the chapter that you do for homework—you have no choice but to go on wild goose chases, most of which lead nowhere. At least nowhere relevant to the solution of the problem at hand.

"But that’s OK. Rarely does it work like this:

• A problem is posed.
• A smart mathematician thinks very hard.
• She solves it!

"More often, what happens is:

• A problem is posed.
• A smart mathematician thinks very hard.
• She gets nowhere.
• Later, sometimes much later, she realizes that "nowhere" actually is
something that a colleague always wanted to know.
• A new problem is solved!"


I love all the stories Paul Zeitz tells about mathematics learning. He is enthusiastic about sharing delight in taking on tough problems--while acknowledging feeling stupid isn't a lot of fun--and encouraging young people to enjoy a sense of adventure and exploration while they learn. I hope that every young person gains opportunities to learn HARD topics that present TOUGH problem-solving situations, so that each learner can take delight in being lost until the learner finds something new, however long that takes. W. W. Sawyer also had encouraging thoughts about what happens to mathematics learners as they outgrow their teachers in their mathematics level. Advanced learners of mathematics may be beyond the teaching level of any teacher in their high school by the time they reach high school age, and much the same happens to advanced music learners as they pursue their study of music. By the late teens, the advanced learners are learning how to learn on their own, while seeking out higher education opportunities that will bring them into a community of experts in their field of study. When a learner who has no teacher at hand to help encounters a tough topic, the learner can still do as Sawyer advised: "We all meet from time to time some particular problem we cannot solve, and we deal with it much as a mediaeval army dealt with an impregnable castle. We go round it and on, no doubt with the hope that it may yield at some time in the future." [Sawyer 1995]

Thinking about Sawyer's image of an army going around a castle reminded me again of Paul Zeitz’s recommendation of Tim Cahill’s line, "The explorer is the one who is lost." And that in turn reminded me that John DeFrancis, author of the first textbook I used for learning Chinese, quoted Robert Louis Stevenson in the preface to the textbook: "To travel hopefully is a better thing than to arrive." When I first read that at age seventeen, I thought that was very off-putting, as the textbook author didn’t even seem to promise that his book would help me learn Chinese. In fact, the book was very helpful indeed, but it was most helpful in building in me an attitude that I couldn’t count on my study of Chinese being easy, as too many of my school lessons had been up till then. As I learned to enjoy the journey, stepping steadily forward day by day to a far-off destination of proficiency in the language, I learned that courage would take me even farther than verbal ability, and my own practice and effort would take me even farther than the best available textbooks--which I had--and miles of audio tapes of the language in the language lab. The same is true of learning mathematics. Mathematicians say that mathematics, like swimming, is not a spectator sport. You have do problems, or get wet, to make progress in mathematics or in swimming. In the study of music, finger exercises and listening to other performers, and playing for other listeners, are all part of steady skill development that may include many frustrating learning plateaus before each new exciting breakthrough in musicality. The main thing is not to be afraid while feeling stupid and lost, but to enjoy the journey and keep stepping forward.

A very interesting popular article about current research on overcoming learning challenges and thriving from them is "The Effort Effect,"

http://www.stanfordalumni.org/news/magazine/2007/marapr/features/dweck.html

which reports the work of psychologist Carol Dweck, who shows how "growth mindset" can make learners smarter. This line of research has been further studied in recent years. Current research suggests that learners who are not afraid to admit that they have more to learn, and who acknowledge they can learn more if they pay attention to their own performance, can enjoy the journey even better than the destination, all the while learning more than learners with less courage in self-examination. A post in the Wired Magazine Frontal Cortex blog by Jonah Lehrer, "Why Do Some People Learn Faster?"

http://www.wired.com/wiredscience/2011/10/why-do-some-people-learn-faster-2/

discusses the importance of not being afraid to face mistakes and learn from them. It is exactly the learners who are most courageous about facing their own mistakes in performance and observing them and correcting them who make the most progress in learning difficult subjects. Lehrer's new book Imagine: How Creativity Works suggests a distinction between problem-solving tasks that yield to steady effort, which Lehrer calls "grit," and problems best solved by insight that comes from going around the problem and letting it rest for a while like Sawyer's medieval army going around a castle.


References:

W. W. Sawyer, Vision in Elementary Mathematics page 5 (original publication by Penguin 1964, reprint by Dover 2003).

W. W. Sawyer, "Catering for the Extremes" Mathematics in School March 1995.