With her book, Knowing and Teaching Elementary Mathematics, Liping Ma caused many to question the depth of teachers' understanding of the subject. For more about Liping Ma, see the related article A Deeper Look at Elementary Mathematics .
by Terese Herrera, ENC Instructional Resources
Liping, through your book we glimpse a much deeper, coherent picture of arithmetic. Our students need that. Our teachers, I believe, are hungry for it. How do we achieve it? I don't know. I really don't know. I think that many should work on this.
First, mathematicians who teach content courses in universities should think about arithmetic in a deeper way. In earlier centuries, great mathematicians devoted their whole lives to solving complex problems through applied arithmetic. Today's mathematicians are working on advanced branches of mathematics, but for those who teach core content for future elementary math teachers, knowledge of advanced mathematics is not enough.
In China, deep arithmetic knowledge is not owned by the mathematicians. It is owned by teachers at the elementary school level. It is not in a book; it is in their minds, and they pass the knowledge on to the next generation. Here in the United States, I personally don't see this body of knowledge in teachers' minds. I don't know why. I hope someone can research and document American teachers' knowledge of arithmetic. This would be very helpful for our teachers.
Math educators need to think not only about how teachers will teach math but also about the math they are going to teach. In the United States, one group of professors teaches content; another group teaches method. But teachers realize that we cannot separate content from method--what to teach from how to teach. In a teacher's real life, they are combined.
We (teacher educators) have no way to put that deep knowledge of elementary math in teachers' minds either. We don't have an answer to give to them. Teachers should know that this is not something that they should wait for other people to give them because we don't have it. We should tell them that it is not their fault at all, and we should encourage them to deepen their knowledge of elementary math through their own teaching.
Teachers themselves should ask "Why?" all the time. When they teach multiplication to their students, they can think about why it works. Then they will learn elementary math in a deeper way through their own teaching of it.
Again, I don't think that I have the answer. I am in the process of looking for answers. There will not be an easy answer for what we should do or a quick solution. We should work together--and work hard--mathematicians, math educators, and teachers.
Your book focuses on arithmetic. How do you see the teaching of geometry in the early grades? Again, for me it is still a question. At the K-6 stage I see arithmetic as the main focus of elementary math, even though elementary geometry should be included.
The geometry that we are teaching now in this country at the elementary level does not connect tightly enough with arithmetic. We teach children the names of shapes--rectangles, triangles, and so on--but not the connection between those shapes. Two congruent triangles can make a rectangle, for example; this is a connection between those figures.
And we need to teach the connection between the geometric figures and other mathematics. For example, we talk about the perimeter of a rectangle as equal to length length width width. The perimeter also equals (length width) 2.
Children could discuss why the formula can be written both ways and which way it makes more sense, is more clear, more mathematically advanced. Doing so, they connect this geometry piece to arithmetic and with the way they think mathematically.
Is there a role for calculators in elementary math classes?
I've had a question in mind for a long time: Since there are calculators that can solve all the skill problems in arithmetic, why do we teach children arithmetic? It is very important to ask this question. It challenges us to sit down and think carefully about what we really want our children to learn.
I believe that arithmetic is much more than skills; it is a basic tool to develop children's minds, their way of thinking. The question becomes: What role does arithmetic play in the development of a child? It is just like asking: What role does literacy/reading play in the development of a child?
I am always asked about calculators in the classroom: Should they be allowed? This is not the question. Calculators are already in use. What we need to consider is how they should be used. In what way can calculators help students learn more mathematics and learn it in a better way?
Your book has influenced the math education community at many levels. What are you working on now? I am a visiting scholar at the Carnegie Foundation because I have a post-doctoral grant for research. My research is on what I call the cornerstone ideas of elementary math. I often hear the terms "core ideas,""big ideas,""important ideas," meaning those ideas that hold all the pieces of a subject. However, it seems that no one is doing systematic research on what these ideas are, why they are central, and how they contribute to the learning of elementary math.
I began last year with pilot studies in kindergarten classes. I developed some teaching materials, including multimedia math games. All of these games are designed for teaching math ideas. My questions are: How do these games work with children? How do students learn? How can teachers learn through teaching?
I go into the classroom to observe the instruction and note what is wrong with my idea. I don't send in others to collect the data. I use myself as the instrument of thinking, because of my early experience in teaching math and observing others teach, and because of my focus on how to create good math teachers. All of those experiences form my vision and make me, myself, an instrument to collect and analyze data. It is a very personal and interior kind of research.
If grants become available, I want to study math textbooks to determine their central ideas and work further with more teachers. The ideas proposed as central to elementary mathematics should then be tested in classrooms, K through 6. We need to lay out the arguments for why certain ideas are central in terms of the subject and in terms of children's learning, and how these ideas can be taught.
I understand you've also written an intervention program for intermediate grades? Houghton-Mifflin wanted to develop a program for children in grades four and up who are below their grade level in math. Cathy Kessel, a fellow researcher who did the heavy editing on my book, and I wrote a student book and teacher's manual. The series is called the Knowing Mathematics Transition Program. Our goal was to create a 12- to 14-week intervention that would bring the students up to their grade level.
Many teachers mentioned that these students don't know the basic facts, multiplication tables being their most obvious trouble. Some teachers let the students use calculators, but using calculators alone is not good. Children need to know multiplication tables on their own. They can push buttons and find the numbers, but they don't know the meaning of the numbers. They might not even know that 35 is two more than 33. Numbers make no sense to them, so we decided to help them to overcome this problem of not knowing basic facts.
We concentrated on the real math behind facts and then the facts themselves. We use a question as the title for each daily lesson: "What is a number?" or "Can you turn addition into subtraction?" or "Do you know the trick for computing with 9?"
Each lesson includes two part--math conversation between the teacher and a group of students, presented as a small play, and a set of math problems. For the lesson on adding with 9, for example, we show this pattern and ask the students to fill in the blanks.
After a few more patterns, we ask them the trick for computing with 9 and why it works. Mainly this lesson is for reflection on the concept. The numbers are very simple, but we want the students to play out the ideas. We think that the easier the numbers are, the easier it is for the students to think about the concept behind them.
For every two-page lesson, one page is a set of problems; children need practice and each problem is closely linked to the day's concept, or to concepts and skills learned in earlier lessons.
The teacher's manual models the classroom discussion for students and teachers, giving teachers questions to ask and explaining why the questions are included. In field tests, both students and teachers enjoyed the discussions. An example of a concept for discussion is the real meaning of subtraction. We usually talk only about "take away," but then why do people call the result of subtraction "difference?" In one lesson we discuss how "difference" is the one concept underlying different models of subtraction. They come to realize that this meaning is much more powerful than "What is left?"
I learned a lot of elementary math through designing this program, through asking myself "Why." Why are addition and subtraction facts so difficult to learn? One consideration is the number of these basic facts: 81 for addition and 81 for subtraction.
The 81 Addition Pairs
What is the real idea behind these pairs? In this table, 20 pairs of numbers have sums less than 10, and most intermediate grade students feel comfortable with these addition facts. There are five pairs that have a sum of 10:
9,1; 8,2; 7,3; 6,4; 5,5
Then there are another 20 pairs with sums beyond 10:
9,2; 9,3; 9,4; 9,5; 9,6; 9,7; 9,8; 9,9
8,3; 8,4; 8,5; 8,6; 8,7; 8,8;
7,4; 7,5; 7,6; 7,7;
Actually, most fourth graders who are behind in addition and subtraction are behind only because they are not good at these 20 pairs.
Now we can concentrate on those pairs, including the eight pairs related to the number 9, which explains why we included the lesson on finding a trick to compute with 9. When students have learned the trick and practiced it, they have learned those eight pairs and can move on to the doubles. They reduce those doubles little by little, ending up with less than ten pairs that have to be learned by rote. Through this approach we teach students basic facts while they learn more about numbers. They cannot learn this by punching the buttons on calculators.
When teachers use this kind of book, where they have to teach math in a different way, we hope they will learn more math naturally through teaching it. They will also learn more about what makes elementary math challenging for students.
What would you say to elementary teachers who want to improve their mathematical knowledge and their teaching? If I were a teacher, I would be very aware that there is no existing body of knowledge that I could study chapter by chapter. If I want to improve myself, I must first depend on myself. I would ask myself "Why?" all the time as I teach elementary math. I would also discuss math teaching and learning with other teachers in my school who teach at the same grade level. Conversation stimulates our thinking.
A layperson only needs to know how to do math, but a teacher needs to learn more because his or her students need a whole picture--a whole concept. Students develop the power of their minds through learning conceptually. Mathematicians may not need these conceptual tools because they have them internalized, but students need tools to develop their abstractive way of thinking.
Teachers who feel that they need more content math knowledge should depend on themselves, never forgetting that they are their own teachers. Their own classroom teaching is the main process through which they can learn. That really makes sense to me.
Could I ask you for any final thoughts? Society is telling teachers so many things, and everybody who tells them something seems to have authority -- psychologists, researchers. Today's workshop says this is good, and tomorrow's workshop tells them something else. Who is the teacher to believe?
Teachers should have confidence in themselves. They are the ones who digest all the knowledge, who listen and change it into a "whole" through their own understanding. I don't know if the word soul is the right word here, but I think teachers are the soul that gives out the light of all the knowledge they hold. I really want them, and us as math educators, to think about that. Otherwise, teachers feel that everybody knows more than they do. Then they lose themselves. I would tell teachers: "Do not forget yourself as a teacher of yourself."
ENC is hosting an online discussion of this article. Join other educators talking about questions such as these:
How can you use Liping Ma's ideas in your teaching? Can you describe a situation in which you used your own class room teaching as a process to help you learn?
Terese Herrera is the mathematics resource specialist at ENC. Her career includes 15 years of teaching mathematics at the middle school and high school levels. Email: firstname.lastname@example.org