Knowing and Teaching Elementary Mathematics

Teachers' Understanding of Fundamental Mathematics in China and the United States

Liping Ma

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367443948
• eBook ISBN: 9781000023084
• Edition: 3
• Topic: Education
• Subtopic: Education General

The 20th anniversary edition of this groundbreaking and bestselling volume offers powerful examples of the mathematics that can develop the thinking of elementary school children.

Studies of teachers in the U.S. often document insufficient subject matter knowledge in mathematics. Yet, these studies give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by reforms in mathematics education. Knowing and Teaching Elementary Mathematics describes the nature and development of the knowledge that elementary teachers need to become accomplished mathematics teachers, and suggests why such knowledge seems more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

Along with the original studies of U.S. and Chinese teachers' mathematical understanding, this 20th anniversary edition includes a new preface and a 2013 journal article by Ma, "A Critique of the Structure of U.S. Elementary School Mathematics" that describe differences in U.S. and Chinese elementary mathematics. These are augmented by a new series editor's introduction and two key journal articles that frame and contextualize this seminal work.

1 Subtraction With Regrouping: Approaches To Teaching A Topic

Scenario

Let’s spend some time thinking about one particular topic that you may work with when you teach, subtraction with regrouping. Look at these questions ( $\begin{array}{cc}52& 91\\ \underset{\_}{-25},& \underset{\_}{-79}\\ & \end{array}$ etc.). How would you approach these problems if you were teaching second grade? What would you say pupils would need to understand or be able to do before they could start learning subtraction with regrouping?

When students first learn about subtraction, they learn to subtract each digit of the subtrahend from its counterpart in the minuend:

$$\begin{array}{c}75\\ \underset{\_}{-12}\\ 63\end{array}$$

To compute this, they simply subtract 2 from 5 and 1 from 7. However, this straightforward strategy does not work all the time. When a digit at a lower place value of the subtrahend is larger than its counterpart in the minuend (e.g., 22 − 14, 162 − 79), students cannot conduct the computation directly. To subtract 49 from 62, they need to learn subtraction with regrouping:

Subtraction, with or without regrouping, is a very early topic anyway. Is a deep understanding of mathematics necessary in order to teach it? Does such a simple topic even involve a deep understanding of mathematics? Would a teacher’s subject matter knowledge make any difference in his or her teaching, and eventually contribute to students’ learning? There is only one answer for all these questions: Yes. Even with such an elementary mathematical topic, the teachers displayed a wide range of subject matter knowledge, which suggests their students had a corresponding range of learning opportunities.

The U.S. Teachers’ Approach: Borrowing Versus Regrouping

Construing the Topic

When discussing their approach to teaching this topic, the U.S. teachers tended to begin with what they expected their students to learn. Nineteen of the 23 U.S. teachers (83%) focused on the procedure of computing. Ms. Fawn, a young teacher who had just finished her first year of teaching, gave a clear explanation of this procedure:

Whereas there is a number like 21 − 9, they would need to know that you cannot subtract 9 from 1, then in turn you have to borrow a 10 from the tens space, and when you borrow that 1, it equals 10, you cross out the 2 that you had, you turn it into a 10, you now have 11 − 9, you do that subtraction problem then you have the 1 left and you bring it down.

These teachers expected their students to learn how to carry out two particular steps: taking 1 ten from the tens place, and changing it into 10 ones. They described the “taking” step as borrowing. By noting the fact that “1 ten equals 10 ones,” they explained the step of “changing.” Here we can see the pedagogic insight of these teachers: Once their students can conduct these two key steps correctly, they will very likely be able to conduct the whole computation correctly.

The remaining four teachers, Tr. Bernadette, Tr. Bridget, Ms. Faith, and Ms. Fleur, however, expected their students to learn more than the computational procedure. They also expected their students to learn the mathematical rationale underlying the algorithm. Their approach emphasized two points: the regrouping underlying the “taking” step and the exchange underlying the “change” step. Tr. Bernadette, an experienced teacher, said:

They have to understand what the number 64 means … I would show that the number 64, and the number 5 tens and 14 ones, equal the 64. I would try to draw the comparison between that because when you are doing the regrouping it is not so much knowing the facts, it is the regrouping part that has to be understood. The regrouping right from the beginning.

Ms. Faith, another teacher at the end of her first year of teaching, indicated that students should understand that what happens in regrouping is the exchange within place values:

They have to understand how exchanges are done … with the base 10 blocks when you reach a certain number—10, in base 10, in the ones column that is the same as, say, 10 ones or 1 ten … they have to get used to the idea that exchanges are made within place values and that it does not alter the value of the number…. Nothing happens to the actual value, but exchanges can be made.

What teachers expected students to know, however, was related to their own knowledge. The teachers who expected students merely to learn the procedure tended to have a procedural understanding. To explain why one needs to “borrow” 1 ten from the tens place, these teachers said, ‘You can’t subtract a bigger number from a smaller number.” They interpreted the “taking” procedure as a matter of one number getting more value from another number, without mentioning that it is a within-number rearrangement:

You can’t subtract a bigger number from a smaller number … You must borrow from the next column because the next column has more in it. (Ms. Fay)

But if you do not have enough ones, you go over to your friend here who has plenty. (Tr. Brady)

“We can’t subtract a bigger number from a smaller one” is a false mathematical statement. Although second graders are not learning how to subtract a bigger number from a smaller number, it does not mean that in mathematical operations one cannot subtract a bigger number from a smaller number. In fact, young students will learn how to subtract a bigger number from a smaller number in the future. Although this advanced skill is not taught in second grade, a student’s future learning should not be confused by emphasizing a misconception.

To treat the two digits of the minuend as two friends, or two neighbors living next door to one another, is mathematically misleading in another way. It suggests that the two digits of the minuend are two independent numbers rather than two parts of one number.

Another misconception suggested by the “borrowing” explanation is that the value of a number does not have to remain constant in computation, but can be changed arbitrarily—if a number is “too small” and needs to be larger for some reason, it can just “borrow” a certain value from another number.

In contrast, the teachers who expected students to understand the rationale underlying the procedure showed that they themselves had a conceptual understanding of it. For example, Tr. Bernadette excluded any of the above misconceptions:

What do you think, the number, the number 64, can we take a number away, 46? Think about that. Does that make sense? If you have a number in the sixties can you take away a number in the forties? OK then, if that makes sense now, then 4 minus 6, are we able to do that? Here is 4, and I will visually show them 4. Take away 6, 1, 2, 3, 4. Not enough. OK, well what can we do? We can go to the other part of the number and take away what we can use, pull it away from the other side, pull it over to our side to help, to help the 4 become 14.

For Tr. Bernadette, the problem 64 − 46 was not, as suggested in the borrowing explanation, two separate processes of 4 − 6 and then 60 − 40. Rather, it was an entire process of “taking away a number in the forties from a number in the sixties.” Moreover, Tr. Bernadette thought that it was not that “you can’t subtract a bigger number from a smaller number,” rather, that the second graders “are not able to do that.” Finally, the solution was that “we go to the other part of the number” (italics added), and “pull it over to our side to help.” The difference between the phrases “other number” and “the other part of the number” is subtle, but the mathematical meanings conveyed are significantly different.

Instructional Techniques: Manipulatives

Teachers’ knowledge of this topic was correlated not only with their expectations about student learning, but also with their teaching approaches. When discussing how they would teach the topic, all except one of the teachers referred to manipulatives. The most popular material was bundles of sticks (popsicle sticks, straws, or other kinds of sticks). Others were beans, money, base 10 blocks, pictures of objects, and games. The teachers said that by providing a “hands-on” experience, manipulatives would facilitate better learning than just “telling”—the way they had been taught.

A good vehicle, however, does not guarantee the right destination. The direction that students go with manipulatives depends largely on the steering of their teacher. The 23 teachers had different ideas that they wanted to get across by using manipulatives. A few teachers simply wanted students to have a “concrete” idea of subtraction. With the problem 52 − 25 for example, Tr. Belinda proposed “to have 52 kids line up and take 25 away and see what happens.” Ms. Florence reported that she would use beans as “dinosaur eggs” which might be interesting for students:

I would have them start some subtraction problems with maybe a picture of 23 things and tell them to cross out 17 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.

Problems like 52 − 25 or 23 − 17 are problems of subtraction with regrouping. However, what students would learn from activities involving manipulatives like taking 25 students away from 52 or taking 17 dinosaur eggs away from 23 is not related to regrouping at all. On the contrary, the use of manipulatives removes the need to regroup. Tr. Barry, another experienced teacher in the procedurally directed group, mentioned using manipulatives to get across the idea that “you need to borrow something.” He said he would bring in quarters and let students change a quarter into two dimes and one nickel:

A good idea might be coins, using money because kids like money…. The idea of taking a quarter even, and changing it to two dimes and a nickel so you can borrow a dime, getting across that idea that you need to borrow something.

There are two difficulties with this idea. First of all, the mathematical problem in Tr. Barry’s representation was 25 − 10, which is not a subtraction with regrouping. Second, Tr. Barry confused borrowing in everyday life—borrowing a dime from a person who has a quarter—with the “borrowing” process in subtraction with regrouping—to regroup the minuend by rearranging within place values. In fact, Tr. Barry’s manipulative would not convey any conceptual understanding of the mathematical topic he was supposed to teach.

Most of the U.S. teachers said they would use manipulatives to help students understand the fact that 1 ten equals 10 ones. In their view, of the two key steps of the procedure, taking and changing, the latter is harder to carry out. Therefore, many teachers wanted to show this part visually or let students have a hands-on experience of the fact that 1 ten is actually 10 ones:

I would give students bundles of popsicle sticks that are wrapped in rubber bands, with 10 in each bundle. And then I’d write a problem on the board, and I would have bundles of sticks, as well, and I would first show them how I would break it apart (italics added), to go through the problem, and then see if they could manage doing the same thing, and then, maybe, after a lot of practice, maybe giving each pair of students a different subtraction problem, and then they could, you know, demonstrate, or give us their answer. Or, have them make up a problem using sticks, breaking them apart and go through it. (Ms. Fiona)

What Ms. Fiona reported was a typical method used by many teachers. Obviously, it is related more to subtraction with regrouping than the methods described by Ms. Florence and Tr. Barry. However, it still appears procedurally focused. Following the teacher’s demonstration, students would practice how to break a bundle of 10 sticks apart and see how it would work in the subtraction problems. Although Ms. Fiona described the computational procedure clearly, she did not describe the underlying mathematical concept at all.

Scholars have noted that in order to promote mathematical understanding, it is necessary that teachers help to make connections between manipulatives and mathematical ideas explicit (Ball, 1992; Driscoll, 1981; Hiebert, 1984; Resnick, 1982; Schram, Nemser, & Ball, 1989). In fact, not every teacher is able to make such a connection. It seems that only the teachers who have a clear understanding of the mathematical ideas included in the topic might be able to play this role. Ms. Faith, the beginning teacher with a conceptual understanding of the topic, said that by “relying heavily upon manipulatives” she would help students to understand “how each one of these bundles is 10, it is 1 ten or 10 ones,” to know that “5 tens and 3 ones is the same as 4 tens and 13 ones,” to learn “the idea of equivalent exchange,” and to talk about “the relationship with the numbers”:

What I would do, from that point, is show how each one of these bundles is 10, it is ...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• A Note about the 20th Anniversary Edition
• Author’s Preface to the 20th Anniversary Edition
• Series Editor’s Introduction to the 20th Anniversary Edition
• Author’s Preface to the 2010 Edition
• Series Editor’s Introduction to the 2010 Edition
• Foreword
• Acknowledgments
• Introduction
• 1. Subtraction With Regrouping: Approaches To Teaching A Topic
• 2. Multidigit Number Multiplication: Dealing With Students’ Mistakes
• 3. Generating Representations: Division By Fractions
• 4. Exploring New Knowledge: The Relationship Between Perimeter And Area
• 5. Teachers’ Subject Matter Knowledge: Profound Understanding Of Fundamental Mathematics
• 6. Profound Understanding Of Fundamental Mathematics: When And How Is It Attained?
• 7. Conclusion
• Appendix
• References
• Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses
• Response to “Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses”
• New to the 20th Anniversary Edition: A Critique of the Structure of U.S. Elementary School Mathematics
• Index