How to Accelerate Mathematics Learning for Elementary Pupils?
I’m a Chinese major myself, so I can take care of the verbal side of my son’s education, but I have been researching math education for about a decade now because of my oldest son’s strong interest in math. Below are my favorite suggestions for parents whose elementary-age children are precocious in mathematics.
1: Get and read a copy of Liping Ma’s book Knowing and Teaching Elementary Mathematics.
You can request it by interlibrary loan if it is not in your local library. Ma’s book makes apparent what kind of foundation is necessary at the beginning for a child to go "as high as he can go" in math. Try solving the teaching problems in that book for yourself, and you’ll see what I mean.
2: Look up the research of Tony Gardiner comparing acceleration to enrichment.
Professor Tony Gardiner in Britain has done extensive research comparing acceleration in the standard school curriculum to enrichment with challenging problems not usually found in the school curriculum as a strategy for preparing bright children for advanced study of mathematics. See, for example,
Mathematical abilities and mathematical skills (.PDF file)
Some Advice for Schools on Provision for the Top 10% in Mathematics
Gardiner has shown what he means in a series of books called Maths Challenge 1, Maths Challenge 2, and Maths Challenge 3, published in Britain. I have bought these books and find they are full of problems to develop a strong understanding of math. The Maths Challenge books are designed by Gardiner for seventh-graders and older students who are in the top 10 percent of the British population. Younger students with good preparation could start using them at a younger age.
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3: Browse Professor Hung-hsi Wu’s website.
and make sure to download and read the draft chapters "Whole Numbers (Draft)" and "Fractions (Draft)" to get more than 100 pages each on those easy
subjects from a thoughtful mathematician with a deep interest in math education. Then read his &How to Prepare Students for Algebra& for more insights.
4: Get and read Taming the Infinite by Ian Stewart.
If your child is an advanced reader Taming the Infinite might be readable by your child solo. Ian Stewart is a professional mathematician who has written several popular books about mathematics. Any of those will show you what your child will be thinking about if he or she takes university-level math courses.
5: Get and read How to Teach Mathematics by Steven G. Krantz.
How to Teach Mathematics is a book pertaining mostly to university-level math study, but with some interesting comments by Krantz on the Saxon math program and on other topics. The book includes essays by other professors of mathematics. Think about what kind of primary and secondary mathematics education (an issue Krantz hardly addresses in his book) would be fit preparation for university study of mathematics.
6: Having done the above, ponder what materials your child learns from.
My top recommendation for a first mathematics program is Miquon Math, a program designed for use over three years that covers almost all of elementary school mathematics from a higher math perspective. For people who have already gone through early elementary math, my number-one recommendation is the Singapore Primary Mathematics series, which is described by many mathematicians as the best mathematics textbook series available in English
an accurate description. I have used the original Singapore third edition of that series with all of my children; the United States edition to which I have linked is also very good. The Singapore Primary Mathematics series is followed by other series from Singapore that take a learner up to all the mathematics needed for A level examinations in the British university entrance system.
Sometimes these two programs, Miquon and Singapore, contain problems that are confusing to American parents who had more conventional math instruction. My friendly suggestion is to take the confusing parts of those books as learning opportunities. Mathematicians linger and ask "why?" and an alternative representation of a mathematical operation (and you can count on the Primary Mathematics series to have accurate representations of mathematical operations in visual, verbal, and other forms) is an opportunity to think about why the content was presented that way. Sure, not every child "gets" the content first from the same kind of presentation, but knowing why all the presentations relate to the same idea is part of understanding mathematics thoroughly.
Many of the illustrations in the Singapore books show examples of manipulatives that could be used in the classroom or at home as a first introduction to a topic. In my house, we usually just went straight to the book, figuring our real life and earlier use of Miquon Math had already provided the "concrete" examples that fit into the Singapore "concrete –> pictorial –> abstract" model of instruction. But the concrete examples are latent in the coursebook (for example, baking cookies, inviting guests to parties, etc.) and may be helpful for many learners.
The Education Program for Gifted Youth (EPGY)
mathematics program is probably wholly unnecessary at the very earliest age level, but it is a great way to move ahead for young people who like that kind of computer-based instruction, especially beginning at about the fourth- or fifth-grade level. EPGY can take learners all the way up to university-level math at their own pace. The ALEKS online program
is a useful supplement to any of the other recommended programs, being almost as good as and a lot less expensive than EPGY, with a fairly complete K-12 mathematics sequence but not continuing to advanced undergraduate mathematics.
Many parents have tried out many other kinds of math programs to help their precocious math learners move ahead with a good foundation. Math programs that I personally do NOT recommend, based on the desirability of a) truly challenging word problems, b) multiple representations of mathematical ideas, and c) clear, CORRECT explanations of mathematical concepts include 1) Saxon Math, 2) Math-U-See, 3) any old, traditional program (e.g., A Beka), 4) exclusive use of "gifted learner" worksheet books or other books that consist mostly of exercise sets, or 5) any "reform" math program used in United States public schools, although the best of these are better than some of the other programs I don’t recommend. These negative recommendations are not intended to offend any parent who has used these programs in a good-faith belief that they are useful math programs, but are mentioned to suggest trying out the truly superior programs if you haven’t done so already.
The very best mathematics textbook series in the world, as best I can ascertain, is the Hua Loo-keng School Mathematics Textbook series published in China. I may have to turn that series into English to give American students an opportunity to learn from the very best. The Hua Loo-keng series makes the curriculum expectations of the EPGY series look like slow learner expectations. The Hua Loo-keng School series doesn’t just go faster but also deeper.
7: Get involved in competition culture
for a reality check on how your child is doing in math.
There is a great variety of mathematics competition programs these days, unlike the days when I went to school, with many programs of differing characteristics. See what local math competitions there are in your area, and what the requirements are for forming a team or joining as an individual. The American Mathematics Competition programs
are readily available worldwide and start at the prealgebra level and go up to qualifying tests for the International Mathematics Olympiad. The better math competitions (MATHCOUNTS
is also in this category) are an excellent reality check on how much math your child knows as second nature.
8: Sign up for talent search tests.
Talent Search Opportunities for 2008
Hoagies’ Gifted: Talent Search Programs
are another way to get a reality check on a child’s math level. The talent search tests are typically a standardized achievement test normed for one age group and given to a younger age group. Most of the regional talent search centers will give you DETAILED information about where your child stands in test performance ranking compared to other children who show up to take the test. The tests vary in format, and thus check whether your curriculum is developing a well-rounded approach to solving (simple) mathematical problems.
9: Live on the Art of Problem Solving (AoPS) website.
The Art of Problem Solving (AoPS) website includes a treasure trove of resources for young math learners, and was long the online home of this FAQ page. The online forum is a great way for young people with interest in math to meet one another and improve their skills. There are many helpful articles on AoPS as well.
I hope this helps! Best wishes to your child.
P.S. Our Experience Base
How We Started out
Your mileage may vary, as the saying goes, but since I advocate a particular approach to math education it’s only fair for me to report on the results I’ve seen so far. To sum up, because my son began saying when he was six years old that he likes math, we have devoted more time to math than many families. But on the other hand we have mostly been fun and CASUAL about math, doing much of the early work orally or with very easy-going attitudes about neatness and rate of errors. We started with the Miquon Math program (we had used a LITTLE of the home activities suggested in the book Teach Your Children Well by Choon Tan, http://www.home.school.nz/bktan.htm but only once in a while). We didn’t always do Miquon Math as interactively as we ought to have, and sometimes my son was just off in a corner by himself, sometimes skipping whole pages in his Miquon Math book. (Our third son was born and I was busy with work while we were using those materials, and sometimes we just didn’t have time to pay a lot of attention to how our oldest was studying then, when we were living in Taiwan.) Sometime in the second year of my son’s formal study, I think before he started the Singapore Primary Mathematics materials, he got a 99th percentile score on a math test, the Woodcock-Johnson individualized achievement test. I think that just reflected that he already knew how to multiply, when many kids of second-grade age cannot multiply. On the same test, he got a 1st percentile score in reading (we hadn’t really taught him to read much yet, but I think that score was unrealistically low). So next I mostly focused on thoroughly teaching my son to read. That didn’t go very well. We found out later my son has strabismus (that is, eyes that don’t point the same direction) and that probably explains, along with living in a non-English-speaking country, why he got a rather slow start in reading. Then he started the Primary Mathematics series, and we began arranging Iowa Test of Basic Skills tests for him once a year at the missionary school in town. On the Iowas, he scored upwards of 80th percentile on math, and downwards of 30th percentile on reading, and that’s when we began to be more concerned about his eyes. (By THEN, around the time of his second Iowa test, I had seen one of his eyes looking SIDEWAYS while his other eye looked at me, so I knew he had a problem.) I think he shed points on math tests at that age mostly because of reading problems. He seemed always able to do the math when he studied with me, but sometimes mistook plus signs for minus signs, for example. We got back to the States and found an optometrist to give our son vision therapy. This was during his fourth grade year, and THEN, at last, he really learned to read comfortably. During that same year my son participated in math competitions http://members.aol.com/mathleague for the first time. Somewhat more than a year ago I got him private IQ and achievement testing, and his math level on the achievement test was again at the 99th percentile, which qualified him for the EPGY correspondence program http://epgy.stanford.edu/ as a way to get him moving faster in math. We were then about in Primary Mathematics 4A, or maybe it was 4B by then. He was right about where a kid from Singapore with his same birthday would have been in school. I like the Singapore program a lot, but I don’t have enough math background to accelerate it, only to teach it at normal speed. (I know other parents locally who have stronger math backgrounds and who know how to speed up that program.) My son complained on the one hand that the Singapore program was "too easy"–that was plausible, based on his test scores–but on the other hand didn’t work independently in that program. I tried out EPGY as a way of getting him moving faster and possibly saving me time for the other kids. What actually happened is that I still had to coach my son a lot on certain aspects of EPGY, but most of the instruction proper was done by the audiovisual "lectures" in the program, and the curriculum pace was much faster. Over last summer he completed EPGY’s fifth-and-sixth-grade course, then the prealgebra course from September through November, and then he started the algebra 1 course in December 2002. He just finished his algebra course last month and got an A. This school year he twice took talent search tests and got 99th percentile scores in math. (The talent search tests show how he performs in comparison with children three years older, and he is at THEIR 98th percentile or so.) My son took the AMC-8 test http://www.unl.edu/amc/e-exams/e4-amc08/amc8.html in November 2002 and scored higher than most eighth graders who showed up to take the test. Just last Wednesday, I had my son sit down with the book 10 Real SATS (Sonlight item number CM111) and take his first practice SAT I test. He got a 640 on his math section, even though he hasn’t learned all the geometry expected on that test yet. I figure I should sign him up for a geometry class soon, maybe the one from Johns Hopkins Center for Talented Youth http://cty.jhu.edu/math/courses/geometry.html and by next April, when he can take an early SAT I in the Midwest Talent Search, he should be ready to score well on a real SAT I taken in real testing circumstances. I give the Miquon Math program a lot of credit for starting out my son right. My wife and I don’t count ourselves as good math teachers–we were liberal arts majors (my wife studied music, and I studied Chinese) who haven’t taken math since high school in the 1970s, finishing up each with a precalculus course that each of us has mostly forgotten. I think the Singapore Primary Mathematics series is a great follow-on to Miquon Math, good for sustaining interest and developing skill without being too boring. My son, still ten years old as I type this, thought the SAT I math sections were "easy," and he made very few mistakes (mostly answer entry format mistakes on the new section with answers the student fills in himself) and skipped very few problems. I would have to say that a ten-year-old who was a nonreader as few as four years ago and who has a legitimate shot at scoring above 700 on the SAT I math section before his twelfth birthday has been served well by his math courses. This is why I think my pet emphases of 1) don’t be too rigid at the beginning, 2) encourage your child to THINK about math, 3) don’t overdo drill, and 4) use multiple materials, not only materials from one company, but always ACCURATE materials are not too far off base. Today, my son writes math problems down, showing his steps, in notebooks without complaint, and he still has his interest in math and curiosity about it intact. He certainly knows his multiplication tables now, but he didn’t memorize them before he had already completed fourth grade. I would encourage parents to relax enough at the beginning to reap the rewards of later forward progress. To answer your specific question, we did the Primary Mathematics series almost entirely orally. But it was near the end of the time when that was our main series (that is, when he was of fourth-grade age) that I began asking my son to use his own paper (sometimes loose-leaf notebook paper, sometimes pads of graph paper) to write down practice exercises or review problem sets, showing his work. That took a bit of getting used to, but of course was a habit he needed to develop by about fourth-grade age. Showing his work neatly on paper will be a big expectation for his next algebra course in the 2003-2004 school year http://www.math.umn.edu/itcep/umtymp/ and that expectation is one of the reasons I am signing him up for that course. It’s possible to insist on too much written work too early–I think Miquon Math strikes a good balance. Where my son will go in mathematics is as yet unknown to me. He may become "only" an engineer, or "merely" a computer programmer, rather than a pure mathematician, but so far every bit of math he has learned is important for his future, whatever he does, and he has had very little wasted time to date in his math education, while still developing an interconnected and deep understanding of the math he has learned. My son is STILL not up to the level of the Hua Loo-keng School Mathematics Textbook series published in China, which for children his age has number theory (modular arithmetic) and other subjects he is wholly unacquainted with. I have one copy of the Hua Loo-keng materials, which are available only in Chinese, for each of the textbooks from first to ninth grade. The first-graders learn Singapore-style block diagrams in that program, and algebraic notation before they leave elementary school, and the junior-high-level books have wonderfully intricate problems that combine algebra and trigonometry to analyze geometric figures, among other treats. The junior-high-level books have been printed in press runs of at least 25,000 copies to date, so there are probably a LOT of kids in China who would think that the SAT I is an easy test by sixth-grade or seventh-grade age. I will have to help my son learn more Chinese (my wife is working on that recently) and yet more math to get him up to "grade level" for that specialized curriculum. But I think he has had about as good a start as a nonmathematician homeschooling dad like me could have given him, and he does still have time to learn what the Chinese kids are learning.
Filed under: Math Blog