CONTENT AND CONDUCT OF THE CLASS
The Prealgebra and Advanced Topics (“Math I” or “Math II”) will follow a syllabus, in development since 2005, that over two years covers most essential topics in
a) a formal course in prealgebra, as it might be taught as an advanced course for gifted elementary pupils or as a review course for adult college students,
b) mathematical problem-solving contests for pupils of late elementary age, with additional topics from upper-division university-level mathematics.
TEXTBOOK USED BY ECAE
The course lays a foundation for further advanced study of mathematics to the highest and most challenging level, and also introduces topics that are interesting and intriguing and not usually taught in the school curriculum. Since the 2011-2012 school year, the course syllabus closely follows the topic order of the textbook Prealgebra (2011) by Richard Rusczyk, David Patrick, and Ravi Boppana. The new book is very thorough and step-by-step in the development of prealgebra topics, and will be especially helpful for students who miss some classes or terms of the PAT course for sports or music program schedule conflicts. It is the specified textbook for both years of the Prealgebra and Advanced Topics course, and is available to ECAE students at a discounted price (currently forty-five dollars, that is $45). Besides the topics covered in the textbook, the course will cover additional topics such as measuring the size of the universe, paradoxes of infinity, coordinate graphing, and other enrichment topics. The students also see problems from Russian-language and Chinese-language textbooks and problems from ancient India, as well as assorted problems from United States problem books for elementary age students and websites for mathematical circle participants and college-age students.
Summer sections of the course (July and August) offer more flexible placement to fit schedule conflicts, as many families use the summer course as a sampler of the school-year course. In all sections at all times of the year, there will be occasional supplemental handouts on problem-solving and advanced topics. During the school year classes, the PAT course is offered in Year 1 (“Math I”) and Year 2 (“Math II”) sections, set for different levels of mathematics background of pupils taking the course. The PAT course is offered generally to pupils of third-, fourth-, or fifth-grade age (broadly defined). If your child is ready for a challenge and willing to work hard in learning how to tackle unfamiliar problems, and practiced in “showing work” while learning mathematics, your child is ready for the course. ECAE has placement procedures for matching learners with the course section that fits their level best if there is any doubt, and students can switch sections up or down if needed to get the best match. On Saturdays during the school year, the 9:00am section will be the Year 2 (“Math II”) section, beginning the fall term with lessons from Chapter 5 of the textbook, working on problems related to algebraic expressions and solving linear equations. On Saturdays during the school year, the 11:00am section will be the Year 1 (“Math I”) section, beginning the fall term with lessons from Chapter 1 of the textbook, working on problems related to how arithmetic works in relation to the field properties of the real numbers. Summer sections of the course (July and August) offer more flexible placement to fit schedule conflicts, as many families use the summer course as a sampler of the school-year course. In all sections at all times of the year, there will be occasional supplemental handouts on problem-solving and advanced topics.
COURSE LOCATION: SOUTH VIEW MIDDLE SCHOOL IN EDINA
The class will be in the usual school-year place, South View Middle School, in Room 135 along the south hallway. Enter the building through the door to the right (east) of the door labeled Door 4, along the south side of the building. PLEASE NOTE that this is not the same location as for summer classes.
South View Middle School, Room 135
The Google Maps link shown above shows the correct parking lot to park in, the South View Middle School south parking lot, accessible from Concord Avenue.
South View Middle School is accessible from the 50th Street Exit on State Highway 100 (for either northbound or southbound traffic) or from the Benton Avenue Exit on Highway 100 (only for southbound traffic) or from various other directions on local streets. The most convenient parking is the South View Middle School south parking lot, accessible from Concord Avenue.
SAMPLE TEXTBOOK PAGES AVAILABLE FOR DOWNLOAD
Lesson 1 for the Year 1 section of Prealgebra and Advanced Topics (Saturday 22 September 2012 11:00am) was on the topic of field properties of addition and multiplication over the integers, with application of those properties to efficient arithmetic. The publisher of the textbook has provided an online excerpt you may download for free
that just happens to include the textbook pages for the first week’s lesson for the Year 1 section.
Lesson 1 for the Year 2 section of Prealgebra and Advanced Topics (Saturday 22 September 2012 9:00am) was on the topics of algebraic expressions and solving linear equations. The publisher of the textbook has provided an online excerpt you may download for free
that happens to cover the second section of the first week’s lesson for the Year 2 section.
Course topics build on basic concepts from the first chapters of the textbook, applying those in all later chapters. Year 1 students learn topics including the real number line, field properties of real numbers and their application to efficient mental arithmetic, squares and other exponents, multiples, divisibility tests, prime numbers, prime factorization, least common multiples, greatest common divisors, fractions, and arithmetic with fractions, equivalent fractions, fraction arithmetic, simplifying algebraic equations, solving linear equations, solving algebra word problems, and decimal arithmetic. Year 2 students learn topics including algebraic expressions, solving linear equations, inequalities, decimals, ratios, proportions, preparation for math contests, conversions, rates, and percents, percentage word problems, squares and square roots, angles, parallel lines, polygons, area, and circles.
The publisher of the textbook has produced a comprehensive series of video lectures complementing the course.
The videos are a good resource for the course, and I encourage students to watch them in advance of each week’s ECAE lessons.
FREQUENTLY ASKED QUESTIONS
Q: What background do students need to start the ECAE math classes?
A: The students mostly simply have to be ready to take on challenging problems. Students beginning the Year 1 section of the Prealgebra and Advanced Topics course will generally be of third grade or fourth grade age, but we have no rigid younger age limit. Students should be comfortable with multidigit addition and multiplication problems, have some familiarity with division and with negative numbers, and some familiarity with adding fractions with like denominators. Students in the Year 2 section of the Prealgebra and Advanced Topics course will generally be of fourth grade or fifth grade age, familiar with factoring integers and fraction arithmetic. Most importantly, any student starting any ECAE math class should be ready to be challenged. The courses are meant to be harder than school courses and to make the students acquainted with how to solve tough problems that are hard to solve at first. The Prealgebra and Advanced Topics course is essentially a prealgebra course spread out over two school years, and with the new textbook it will include many HARD problems.
Q: Is homework required for ECAE courses?
A: ECAE is not in the business of providing students letter grades, so students are expected to be self-motivated and eager to learn mathematics. The PAT textbook provides exercises, practice problems, and challenge problems, and the instructor provides additional in-class problems for group solution by discussion or individual work at the whiteboard, as well as handouts with other exercises and problems and math contests from the Continental Mathematics League (CML) and Math Olympiads for Elementary and Middle Schools (MOEMS). Much learning and practice will happen in class; students will have lots of opportunity to practice and learn outside of class too with the course materials, as each family decides. I am always happy to discuss with students any problem they have worked on outside of class–even if the problem is not from the ECAE math course.
The Prealgebra and Advanced Topics course is structured so that students can start in the middle, or continue from the beginning, or attend some terms and not others as schedule conflicts demand. I will orient everyone to the course at the beginning of the first class. Some students are returning students and some are new to the course.
Parents are very welcome to stay in the classroom or outside the classroom or somewhere nearby as they wish. The typical conduct of the class includes having problems on the whiteboards as the students arrive, and for students to take turns going to the whiteboard to work problems. We discuss problem-solving approaches and different methods of solving each problem interactively during the class. Some parents have told me that they especially value the way each learner learns new problem-solving approaches from the other participants in the class.
NO SNACKS DURING CLASS
With the concerns of students who have food allergies in mind, we will no longer have the custom of having a snack break during the class. Water is welcome in the classroom, if it’s in a container that prevents spills. Please leave food in your car or farther away from the classroom.
PROBLEMS VERSUS EXERCISES, REPETITION, AND HOW TO LEARN MATH
Below (labeled LONG APPENDIX) are some materials I’ve prepared as Frequently Asked Question files to answer commonplace parental questions about appropriate math education for highly able students. The first FAQ is about the distinction between “problems” and “exercises.” In ECAE classes, we try to serve up lots of problems. The second FAQ is about how much repetition in a lesson is too much–and the answer to that is that it depends on whether the learner is repeating exercises, or repeating problems. The third FAQ includes links to online articles and quotations from printed books and articles describing why the ECAE math classes are organized as they are.
QUESTIONS OR CONCERNS
As always, if you have questions or concerns, feel free to contact me. I love to hear from parents of pupils in my classes or from the pupils themselves. By the way, I encourage you to use voice telephone calls to reach me if that is convenient to you. I read in the summer of 2011 in the New York Times that some Americans rarely make voice phone calls anymore, but I am still glad to receive telephone calls, so feel free to phone me for a conversation about your child’s experiences in the course or for any questions you have.
(952) 200-3146 (cell)
1) PROBLEMS VERSUS EXERCISES
I frequently encounter discussions among parents about repetitive school mathematics lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between mathematics exercises (good in sufficient but not excessive amount) and mathematics problems (always good in any amount).
EXECUTIVE SUMMARY: What makes the difference between an “exercise” and a “problem” is how readily a learner can figure out how to begin when faced with the exercise or problem. All learners need some exercises at all stages of learning, but it is possible to go wrong by having too few problems in the learning process. Basically one can never solve too many problems.
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.
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 tackling 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
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)
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.
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.
“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. Similarly, mathematician Terence Tao, who is both a gold medalist in the International Mathematical Olympiad (the youngest ever) and a Fields medalist for his contributions to mathematical research, says “Learn and relearn your field,”
and “Ask yourself dumb questions – and answer them!”
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,”
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.
Another review of that excellent book by mathematician Roger Howe
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.
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”:
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.
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.
Timothy Gowers, a mathematician who is both an International Mathematical Olympiad gold medalist and a Fields Medal winner, wrote “The Two Cultures of Mathematics”
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?”
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.”