Customer Reviews

Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)

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I am working on certification in secondary mathematics. This one book has given me more insight into what is wrong with mathematics education in the USA and what needs to be done than anything else I have read or discussed in class.

The author's key point is that even the best elementary school math teachers in this country have only a shallow, cookbook knowledge of arithmetic and are not trained to think mathematicaly.

One consequence is that the emphasis in mathematics teacher training on new instructional practices: use of manipulatives, "authentic assessment" collaborative learning, etc. is at best misplaced.

There is much interesting information on Chinese educational practices. Math at all levels is taught by specialists who have only the equivalent of a Chinese high school education. Classes are very large but teachers have about an hour of time for preparation, grading homework, and student conferences for every hour of instruction. Chinese math teachers spend many, many hours working with the curriculum as learners both individually and in groups.

The book is a rich source of ideas that might be adapted to the American environment to improve math instruction.

Elementary school teachers are expected to teach almost everything: math, reading, science, social studies, and writing; along with nurturing, soothing, and encouraging. It's not an easy job. It's also hard to be an expert in any one piece of the job. But now, many are hearing that we're losing the "math race" to other countries. The drums of "teacher competency" are booming... and any wise teacher knows where the drum sticks will be landing next!

Liping Ma's book comes at an opportune time for those teachers and should be read by all. It dives into a central problem that elementary teachers face when we consider improving our math programs: How could going off and learning more math help, for instance, in a 4th grade fractions unit? Furthermore, having, typically, been taught mathematics, ourselves, as a process of memorizing and applying procedures, we often teach it that way as well, thinking "how much more can I study the `flip and multiply' rule for fraction division?"

This book answers those and many other questions, while opening many new ones. There's more to math, even "kids math" than meets the eye.

Ma demonstrates that American teachers do not necessarily suffer from a lack of breadth or extensiveness of mathematical training. Adding more `higher math' to our training really would not help us teach arithmetic. We lack deep knowledge of "fundamental mathematics." Ma's claim is that what we need to do is to dig deeper into the underpinnings of "elementary" math - to discover that there is much more to understand about such fundamental concepts. There really is much more to subtraction than remembering when to "regroup." Division of fractions actually represents two or three fundamentally different processes which, confused, can be at the center of students (and teachers) uncertainty. Imagine that!

These are thought provoking ideas, well presented. In the face of a growing national debate about "competency", we would do well to add this element to the discussion of math teaching. Otherwise, the politicians will provide their inevitably simplistic answers.

With exceptional clarity, Ma compares American and Chinese teachers by discussing their responses to four teaching situations. The Chinese teachers, despite less formal education, have a much deeper understanding of the elementary mathematics they are teaching. Ma explores the components of what she calls "profound understanding of fundamental mathematics," and also the professional conditions that encourage it. Highly recommended for anyone involved in the preparation or professional development of teachers. Also highly recommended for educational policy makers.

Interviews teachers and displays the difference between procedural understanding of math and conceptual understanding of math. Delves into how and why US and Chinese teaching of math are so different.

An American teacher with only a procedural understanding said this about teaching regrouping with manipulatives:

"I would have them start some subtraction problems with maybe a picture of 23 things and tell them to cross out 17 things and then count how many are left. . .. . .I might have them do some things with dinosaur eggs, or something that would sort of have a little more meaning to them. Maybe have them do some concrete subtraction with dinosaur eggs, maybe using beans as the dinosaur eggs or something."

What? Dinosaurs are the key to effective teaching of math? This approach does not explain why we regroup! It does not even touch on place value. You have got to read this book to believe what goes on in way too many American classrooms!

An American teacher with a conceptual understanding of math had a much better way to use manipulatives in teaching regrouping. She used single sticks and bundles of ten sticks to show the mathematical principle of equality. She said she would stress that when you have 53 sticks, the total is still 53 sticks whether arranged in 5 bundles of ten, plus three sticks; or 4 bundles of ten, plus 13 sticks. THIS is a manipulative approach that actually works to teach the concept of regrouping because it draws on the fundamentals of math. One has to demonstrate to the children the idea that you can change the FORM of the number without changing the number itself.

Watch out for page 31! You may CRY when you see that many American elementary teachers don't recognize the implied zeroes in multi-digit multiplication. One actually said you could use apples, oranges, or even ELEPHANTS to help you remember to move the columns over. Fruit? Animals? What about TENS? This experienced math teacher did not know that 237 shifted over a column stands for 237 tens. Chinese math teachers would not think of dressing up the process with fruit and animals. They simply teach the mathematical principles behind the procedure.

Parents and teachers alike need to gobble this book right up, right now, despite its high price. We must work towards more conceptual understanding of math by elementary school teachers or we shall never climb out of the morass. Buy a copy and share it around---it is a crucial work.

This is education research at its best. It directly points the way to better classroom preformance, which is ultimately what education research is about. It gives a very persuasive argument that good mathematics teaching must start with a total command of the relevant mathematics. Its implications on the professional development of mathematics teachers should be taken to heart by educators, state agencies, and administrators.

Liping Ma has shown us what it means to know elementary school mathematics deeply, and has suggestions about what can be done to help our teachers acquire this knowledge. Read and share this book with anyone who cares about the education of elementary school children.

This book provides a starting place for an important and long overdue discussion. Teachers, mathematicians and preservice programs could come together to develop professional development programs around this one book. Understanding our own misunderstandings in mathematics will help to change instructional practice. Every school should have at least one copy.

For many years, I have heard about how Asian countries typically outperform America in tests relating to concepts in mathematics. I've often wondered why; I thought perhaps it's because testing is more rigorous in these countries or perhaps the students have more homework and are able to practice math more frequently. However, after reading this book, it appears the teaching method may make the biggest difference. Better teachers have probably had better experiences of being taught when they were students. This cycle continues on, therefore producing more competent students and perhaps fostering an interest in teaching in the future.
Much of the data contained in this book is based upon interviewing 23 American elementary-level teachers and 72 Chinese elementary-level teachers in regards to teaching processes. Each chapter deals with a different scenario: subtraction with regrouping, multi-digit multiplication, division by fractions, and relationship between perimeter and area. The point of each topic analysis is to see how the different teachers responded to each of these concepts.How would they teach it? Would they provide additional examples outside of the textbook? How would they correct a students mistake? I found Chapter 4's scenario particularly interesting: how would a teacher prove (or disprove) a student's unconventional new theory about area and perimeter?
Many excerpts from the teacher interviews are provided to show the differences between the understanding of the U.S. teachers and the Chinese teachers. The biggest problem with the American teachers that I noticed was that a fairly large majority of them were usually unsure of the rationale of why particular math concepts were carried out a certain way. Even when the math procedures were carried out correctly, the American teachers seemed much more unsure of their own knowledge than the Chinese teachers. The Chinese teachers usually provided more concise responses in regards to teaching the different types of material, as well as providing alternate ways to approach the problem.
The Chinese teachers most often showed a very deep understanding of the topic at hand, going above and beyond any of the American teachers' explanations. The most interesting observation is made by one of the teachers in Chapter 1; Tr. Chen mentions: "The operation of subtraction with decomposition (the Chinese term for subtraction with regrouping) is the application of several ideas rather than a single one. It is a package, rather than a sequence, of knowledge.". This "knowledge package" term is mentioned several times later on in this book, and I think it is an incredibly important indication of the difference between Chinese and American teaching methods. Every subject in mathematics, at the elementary level and higher levels, can be related to other important operations. For example, subtraction with regrouping should not be taught alone, as if it isn't relation to other concepts; it should be related to subtraction without regrouping, addition and subtraction as inverse operations, and so on. Many more examples of knowledge package diagrams are given in the book.
In conclusion, I find that "Knowing and Teaching Elementary Mathematics" would be a helpful book to read for anyone interested in wanting to improve their method of learning or teaching mathematics. For that matter, many of the techniques employed by the Chinese teachers interviewed could be applied to other subjects as well; integration of how one topic is directly related to others in a "knowledge package" is incredibly helpful in better understanding any subject.

Never did I think that I would be interested... heck... excited... about a math book... but this is a great book, and good reading for anyone interested in education... It has given me a great deal to think about as a EFL teacher, a field where most "schooling" consists of teaching procedurally...
This excerpt from an editorial review of an entirely different book (The Teaching Gap) but summarizes Liping Ma's research nicely... "American teachers... tend to emphasize terms and procedures, thinking of math as a set of tedious skills... In contrast, [Chinese] teachers are more likely to emphasize ideas, expecting the concepts alone to stir students' natural curiosity."
It doesn't matter what field you are teaching, teaching a "procedural understanding" is like giving the students a fish; it will feed them for a day... a "conceptual understanding" will feed them for a lifetime...

That's right: a teacher (undoubtedly an American teacher), and even worse, one working on a master's degree in education. And yet, she (to pick a gender to use) is unable to see any worth in this great book. Furthermore, her review illustrates one of the problems with many teachers today in the US: the lack of critical thinking skills, not to mention an apparent ignorance about research methodologies.

That reviewer complained that the author "neglects to even mention any cultural differences between the two societies that she is comparing" and saw that failure as a fallacy that undermined everything said in the entire book. How cultural differences would alter any of the valuable lessons that we can learn from this book about a better way of teaching mathematics in elementary school and the importance of teachers having an in-depth knowledge of the subjects that they are teaching is completely lost on me. Maybe that's because they would not make any difference!

Math is a universal language, and we should be able to learn something from other cultures who we already know are producing children who consistently outscore American children by a wide margin.

And guess what: one of the basic goals of cross-cultural studies is to find out if others are doing something better and, if so, whether their methods can be applied successfully to help us do something, like teaching mathematics, better. Of course there are some cultural differences between the US and China, but so what? The point is to examine other societies who are apparently doing a better job in some area and see what they are doing differently and what can we learn from them.

The fact that a teacher found this book "boring and not worth the paper that it is printed on" should tell those of you not working in the field of education how bad it already is in our schools and how much worse it is probably going to get in the future. I teach remedial mathematics at the college level. In case any of you reading my comments did not know it, many of our schools are, and for some time now have been, graduating kids from high school who cannot add and subtract integers because they do not understand the most basic facts about our number system and how it works. None of my students have even heard the term "base 10," which is what the "composing" and "decomposing" concepts used by the Chinese teachers relies so heavily upon, and is what determines the value of every number in the system that we use in everyday life (computers, of course, use base 2).

Is there any hope for improvement in the future? Frankly, I don't think so because we just keep throwing money at the problem (giving it to the same people who helped cause the problems), rather than doing those things that are necessary to bring about meaningful changes.