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Concrete Experiences Do Not Automatically Lead to Understanding and Generalization

by Jinfa Cai
November 2001 jinfa 

Previous cross-national studies have consistently shown that the mathematics performance of students in the United States is not as high as expected or desired, especially when compared with the performance of students in certain Asian countries. As a result, some people have argued for drastic reform of mathematics education in the United States. However, other people have asked why we should care about the international rank for students in the United States in mathematics, since, after all, the United States is still the greatest country economically and academically. For me, proponents of both these positions seem to be overlooking a more important question: What can we learn from cross-national comparative studies in mathematics to improve students' learning? One thing that these results tell us is that students in the United States rely on concrete strategies and experiences to solve mathematics problems more frequently than their Chinese counterparts do. Let me elaborate.

It is generally accepted that students in the United States do not perform as well as Asian students on tasks that require them to apply mathematical knowledge and skills routinely learned in school. The good news is that several recent cross-national studies have shown that for tasks assessing complex problem solving, the differences between students' performance in the United States and in Asia are less pronounced. In fact, students in the United States have similar or even better rates of success than Asian students in solving some problems.

The staircase problem, in figure 1, and the doorbell problem, in figure 2, were used in a research study, "Mathematical Thinking Involved in U.S. and Chinese Students' Solving Process-Constrained and Process-Open Problems," by Jinfa Cai (Mathematical Thinking and Learning: An International Journal 2 (2000): 309–40). In solving the "staircase problem," shown in figure 1, sixth graders' success rates in the United States are about the same as those of Chinese sixth graders. However, closer inspection of their explanations reveals interesting differences in how students from the two countries attempted to solve this problem. Sixth-grade students in the United States tended to use concrete problem-solving strategies, whereas Chinese students tended to use generalized problem-solving strategies.

 

Percents of U.S. and Chinese Sixth-Grade Students
with Correct Answers for Each Part of the Staircase Problem

Look at the figures below.

figure 
step  percent  blocks 

1 step

2 steps 3 steps 4 steps
  1. How many blocks are needed to build a staircase of five steps? Explain how you found your answer.

    United States: 91%; China: 92%

     

  2. How many blocks are needed to build a staircase of 20 steps? Explain how you found your answer.

     

    United States: 28%; China: 22%

Fig. 1 

To illustrate, when students were asked to find the number of blocks needed to build a 20-step staircase, about a quarter of the sixth graders in the United States attempted to actually draw a 20-step staircase to arrive at an answer. If a student is able to draw a 20-step staircase, the student may know the pattern and how each figure in the pattern changes from one step to the next. However, such drawing strategies are cumbersome and inefficient, and fail the student when the number of steps is large, such as in the 100-stair problem. By contrast, only a few Chinese sixth graders solved these problems in that way—most tried to develop a "closed-end" formula for the number of blocks needed, a formula in which each step could be calculated independently of the number of blocks needed for the previous stairstep.

Similarly, in solving the "doorbell problem," shown in figure 2, a considerable number of students in the United States used concrete strategies in their explanations, whereas many Chinese students used abstract strategies. Using a concrete strategy, students in the United States made a table or a list or noticed that each time the doorbell rang, two more guests entered than on the previous ring, and students actually added 2's sequentially to answer the questions—an iterative process. Using an abstract, or closed-end, strategy, the successful Chinese students noticed that the number of guests who entered on a particular ring of the doorbell equaled twice that ring number minus one (that is, y = 2n – 1, where y represents the number of guests and n represents the ring number). Others noticed that the number of guests who entered on a particular ring equaled the ring number plus the ring number minus one (that is, y = n + (n – 1)). Although students in the United States were better than the Chinese students at finding the number of guests who entered on the tenth ring and got credit for a concrete explanation in Part B, they did not perform quite as well as the Chinese students on the Part C of this problem—if 99 guests entered, what ring was it? This disparity appears to have occurred because more Chinese students than students in the United States used abstract, or closed-end, strategies.

 

Percents of U.S. and Chinese Sixth-Grade Students
with Correct Answers for the Doorbell Problem

Doorbell Problem

 

Sally is having a party.

The first time the doorbell rings, one guest enters.
The second time the doorbell rings, three guests enter.
The third time the doorbell rings, five guests enter.
The fourth time the doorbell rings, seven guests enter.
Keep going in the same way. On each ring, a group enters that has two more persons than the group that entered on the previous ring.

  1. How many guests will enter on the tenth ring? Explain or show how you found your answer.

    United States: 72%; China: 58%

     

     

  2. In the space below, write a rule or describe in words how to find the number of guests that entered on each ring.

     

    United States: 70%; China: 53%

     

  3. 99 guests entered on one of the rings. What ring was it? Explain or show how you found your answer.

     

    United States: 22%; China: 27%

Fig. 2 

Why were students in the United States less likely than Chinese students to use generalized problem-solving strategies? One possible answer is that teachers in the United States less frequently encourage their students at this level to move to more abstract representations and strategies in their classroom instruction. One of the common conceptions held by some teachers in the United States is that concrete representations or manipulatives are the basis for all learning. These teachers believe that pictorial representations or concrete materials can facilitate students' conceptual understanding. However, some research shows that the use of manipulatives or concrete experiences alone do not guarantee students' conceptual understanding. The purpose of using concrete visual representations is to enhance students' conceptual understanding of the abstract nature of mathematics, but concrete experiences do not automatically lead to generalization and conceptual understanding.

Making a list is a viable strategy to find the ring number when ten guests enter, but such a concrete strategy, if not extended to the abstract level, may impose limitations on students' development of mathematical-reasoning abilities. Thus, students should be given the opportunity to construct their own representation of mathematical concepts, rules, and relationships. However, we should expect them to have an understanding that goes beyond "concreteness." For example, teachers may start with concrete manipulatives to encourage students to use their own strategies for solving problems and making sense of mathematics. But after that, teachers might gradually encourage students to develop more efficient and generalized solution strategies.
 

 

Jinfa Cai is an associate professor and director of secondary mathematics education at the University of Delaware. He is interested in how students learn mathematics and solve problems, as well as how teachers can provide and create learning environments so that students can make sense of mathematics. He has explored these questions in various educational contexts, both within and across nations. He is on the Editorial Panel of the Journal for Research in Mathematics Education and on the Executive Board of Special Interest Group of Research in Mathematics Education.
 

 

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