Elementary math is anything but "basic" —
For Homeschool Teachers: by Jennifer Dees One of the first questions you're likely to see from a new homeschooling parent on a homeschool mailing list is this: "What curriculum should I buy for math?"
Plenty of advice is sure to follow, but the best answer might be: This should not be your first question. Instead, there are some questions to ask of yourself. What do you know about math? How do you feel about math? These factors are likely to determine how successful you will be at helping your children learn math. And if you spend some time learning a bit about mathematics teaching, you will then have your own very good ideas about what you want in mathematics learning materials. This doesn't mean you have to go back to college and major in math. Most elementary teachers didn't, and didn't even take many math courses. According to a paper from New Horizons for Learning, "Most elementary math teachers have little preparation beyond college algebra and a two-credit teacher education course in math methods or about twenty-four classroom hours. Middle school math teachers frequently have a major in a field other than math." In fact, American schools generally don't do a great job of helping children learn math, with only a third of students even getting to high school algebra, and fewer still going on to more advanced math. So why would we homeschoolers emulate what schools do, when we have the opportunity to be so much more imaginative? Why would we buy a math curriculum that consists of workbook problems to be completed at the kitchen table? Perhaps because that is the way we learned, and the only way we know to "do math." Perhaps our understanding of math is not strong enough that we can be more creative about it, as we might be in other areas. Or maybe, like the majority of Americans, we're even a little afraid of it, or have what is called "math anxiety." If we are, our children will surely feel it, and pick up on it themselves.
This article was inspired by an important book, Knowing and Teaching Elementary Mathematics, by Liping Ma. A book that has revolutionized mathematics education of teachers, it shows why American students underperform in math, and what can be done about it. Across the U.S. and in Canada, this book has sparked changes in colleges that prepare new elementary education teachers to teach math. And it has important implications for homeschool parents who are helping their children learn math, as well. I believe this book ought to be required reading for any new homeschooling parent who will be helping his or her children learn math. If you read this book, it will profoundly affect the math materials you'll want to choose for your children, as well as your ability to help them learn. I'll explain why I think this book is so important, and then discuss some materials that will help you follow up on its recommendations. Knowing and Teaching Elementary Mathematics "Chinese students typically outperform U.S. students on international comparisons of mathematics competency," writes math education researcher Liping Ma. This is true at all levels, but it's especially evident at the advanced end. Almost all Chinese students take high school calculus, but only 13 percent of American students do.
Liping Ma grew up in China, still in the midst of its Cultural Revolution. As an eighth grader, she was sent from Shanghai to a rural village for "cultural reeducation." Soon she was asked to teach at an elementary school. Over the next seven years, she taught all five grades. Later, she returned to the city and earned a master's degree. Then she came to the U.S. for further graduate work. She studied research on elementary math teachers here and found their methods and abilities to be very different from the Chinese teachers she had known. In a unique position to compare math teaching of Chinese and American teachers, she did further research and developed her findings in a dissertation for her Ph.D. at Stanford. Her advisers and other mathematicians found it so compelling that she was urged to turn it into her very accessible and readable 1999 book, Knowing and Teaching Elementary Mathematics. Why the differences in math performance? Ma found differences that were much more intriguing than just hours spent studying, as is sometimes reported. She found that Chinese teachers had a much greater conceptual understanding of elementary math, and that they taught elementary math in conceptual ways. In contrast, she found that American teachers were overwhelmingly focused on procedures and algorithms, and often were unable to explain the concepts underlying those procedures. This was especially true at the higher levels of elementary math. For example, Ma asked both Chinese and American teachers to solve the problem, 1¾ ÷ ½. All of the Chinese teachers could calculate the answer, while only 43 percent of the American teachers could. Even worse, when asked to explain the meaning of division by fractions in a worded example, only one of 24 American teachers was able to do so, whereas 90 percent of the Chinese teachers did so, and most offered several different kinds of examples. One of the most fascinating aspects of the research Ma reports is in the verbal explanations teachers were asked to give. She actually had an American teacher explain that you would do a procedure a certain way "...because that's the way I was taught to do it." I'm an American, and the last thing I want to say is that these teachers were less capable than their Chinese counterparts because they are American. Instead, I believe the procedural focus without conceptual understanding that Ma observed among American teachers is due to "the way they were taught to do it." That's the way I've always heard it should be Generations of American students have been taught to execute algorithms, and to do it quickly. In the book The Mathematical Education of Teachers, the authors note, "Public education in the United States has historically had a utilitarian focus, which in mathematics, emphasized arithmetic skills and problems from commerce, such as compound interest. Until recently, only high school students in college preparatory tracks studied algebra, and then often just for one year." The big problem with a strictly procedural approach is that even students who complete procedures adequately may be at a loss later, when they encounter advanced mathematics that requires more conceptual understanding. Ma refers to the inadequate elementary math taught currently as "shopkeeper arithmetic." Students are still learning elementary mathematics in a computation-intensive way. But students will grow up to perform jobs in a world in which cash registers compute change, and computers track banking and commercial transactions, along with everything else that can be calculated. Yet if anything, the incessant testing brought on by No Child Left Behind has only intensified the effort to drill in procedures, with no time available for exploration and discovery of concepts. Exploration and discovery are key ways Chinese teachers help their students learn conceptually, Ma notes. Students are given ample opportunity to work together, discuss possible solutions, and speculate about theories. They are encouraged to look at many possible ways to solve a problem, not just find one "correct" one. One of Ma's research questions looks at how Chinese and American teachers respond to a novel idea raised by a student. A student speculates that as perimeter of a rectangle increases, area will also increase. Chinese teachers said that they would encourage students to examine this idea on their own, and to explore it mathematically, to try to find out whether it is true or not. In cases like this, they use class discussion about mathematical ideas. Most American teachers assumed it must be true, or thought it might be, and said they would have to look it up in a book and then tell the students what they found. (Perimeter and area are not related.)
Chinese teachers tended to "justify" their explanations mathematically, citing the distributive, commutative, and associative properties and other such rules. They tended to think mathematically in terms of proof. Ma found that some Chinese teachers had what she calls Profound Understanding of Fundamental Mathematics; none of the American teachers had this. With it, she says, teachers understand the connectedness of underlying concepts. She argues that without a teacher having such comprehensive understanding, students have little hope of attaining such an understanding themselves. And she examines how and when Chinese teachers gained this understanding. She found that even before they became teachers, almost all the Chinese were more mathematically capable and had a greater conceptual understanding than the American teachers, because that is the way they learned math as children. But those who had the profound understanding seemed to gain this after they had become teachers. Learning to teach, teaching to learn Ma found that the Chinese elementary teachers "study the teaching materials intensively." Their teaching manuals include a discussion of each topic focusing on these questions:
She also found that the Chinese teachers learn mathematics from their colleagues. Once weekly, they get together formally for an hour to share their ideas and reflections on teaching. The teachers also told Ma that they learn from their students. When students propose novel ways of looking at a problem, the teachers tend to listen to them and give them time to make their cases. Teachers also said that they learn mathematics by doing it, such as trying on their own to solve a problem several different ways. Says Ma in her conclusions, "In the United States, it is widely accepted that elementary mathematics is 'basic,' superficial, and commonly understood. The data in this book explode that myth. Elementary mathematics is not superficial at all, and anyone who teaches it has to study it hard in order to understand it in a comprehensive way." So where do we go from here? Should this be scary to homeschool parents who haven't attended a teacher training program in mathematics? I don't think so. Just as, unfortunately, teachers in most U.S. schools are on their own to try to improve their skills in mathematics teaching, homeschool teachers can also undertake to increase their understanding of mathematical concepts and how to teach them to their children. Some of Ma's recommendations for teachers can apply to homeschool as well as institutional teachers. She suggests that teacher knowledge and student learning can and should be addressed at the same time. Work on each, she says, should support the improvement of the other, as they are interdependent processes. As homeschoolers, we can learn together with our children, as long as we have an interactive, open relationship in which the child's exploration and discovery are encouraged. If you really want to know something, teach it. (I remember my mother telling me that she never really understood calculus fully until she taught it at the college level.) Just be sure to give your children a chance to do some teaching, too, so they'll get the highest-quality learning experience. Just what are all the concepts of elementary math you need to know to help your children learn math conceptually? For one example, Ma's book itself was terrifically helpful to me in focusing on place value in our number system as an underlying concept when helping my daughter learn multi-digit addition and subtraction. Ma illustrates using place value and powers of 10 to explain "decomposing" a ten to make 10 ones and the inverse, composing a ten from 10 ones. This replaces the confusing "borrowing a ten" or "carrying a ten" used by Americans. Once a child has a strong understanding of place value, the "composing" and "decomposing" terminology for multi-digit operations makes sense conceptually. Ma's book and other research studies were wakeup calls to the mathematics education establishment. Since her book was released, some materials have been developed to help address the issues raised. One excellent book, The Mathematical Education of Teachers (2001), developed by the Conference Board of Mathematical Sciences, provides a complete overview of what teachers from elementary through high school level math need to know. This is very accessible material for homeschool teachers, and is available in its entirety on the Web, at no charge. The teacher manuals that accompany Singapore Math materials are very good in their explanations of concepts covered in their textbooks and workbooks. Since their textbooks have very little verbiage addressed to children, it would be advisable to review both the student's books and the teacher's manual for that section before presenting it to your child, if you have any doubt about the concepts being presented. Talking with your child extensively as he works with math materials is preferable to leaving him to work through them on his own. Even if he is capable of completing the work on his own, the opportunities to explore concepts and attain deeper understanding are lost if math consists only of pages of problems being completed. If you feel at all shaky about your own conceptual understanding of elementary mathematics, an excellent book for teachers is Mathematics: A Good Beginning, by Troutman and Lichtenberg, described as encompassing "...a wide range of topics that prospective elementary and middle school teachers need to know to present mathematics lessons." There are, of course, many excellent Web sites for teachers that show how to teach particular concepts. One example is IMAGES, Improving Measurement and Geometry in Elementary Schools. Sites like this are intended for schoolteachers but are excellent resources for homeschool teachers as well. The fantastic Living Math site is custom-tailored for homeschoolers. Published by homeschool mom (and former CPA) Julie Brennan, Living Math provides an entire page of links to readers for learning specific math concepts. The rest of the site is an invaluable resource as well. The Math Forum at Drexel has a fabulous tag line: "People learning math together." It's a great content resource and that's a great goal for all of us as we help our children build a conceptual foundation for their mathematical understanding. Jennifer Dees taught middle school math and algebra in 1981-1982, her first job after graduating from college. She then went on to an entirely different career, founding a computer magazine and serving as its editor and publisher for 9 years. When she began homeschooling her daughter years later, she found she had much to learn about teaching math. She continues to draw inspiration and guidance from her mother, Roberta Dees, a retired math and math education professor, and a gifted teacher. For more information Knowing and Teaching Elementary Mathematics, by Liping Ma, 1999. Reviews of Ma's book
What you need to know to teach elementary math
The problem with mathematics learning in the U.S. today
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© 2006 Jennifer Dees |
