**[Here is another essay on Study from 1996.]**

**[Here is another essay on Study from 1996.]**

A “blank” is created in the mind by a **“concept not fully understood.”** Such blanks prevent later concepts from being understood and, thus, multiply themselves rapidly. In trouble-shooting, if a student is unable to grasp a concept, it is certain that there is an earlier concept not fully understood. The following case demonstrates the effect of earlier “blanks.”

Once, a mother came to the Math Club with her daughter who was in fifth grade. The daughter was having great difficulty in math. According to her mother, she did not want to memorize the multiplication tables, and that was the problem. The troubleshooting went something like this:

### TUTOR: “Is there something in math you don’t feel quite

### comfortable with?”

### STUDENT: “Yes… multiplication.”

### TUTOR: “Alright. What does the word MULTIPLY mean?”

### STUDENT: “Umm…”

(The tutor explained the process of multiplication as “repeated addition.”)

### TUTOR: “I am going to check you out on the multiplication of

### two single-digit numbers. What is three times two?”

### STUDENT: “Six.”

### TUTOR: “What is four times three?”

### STUDENT: “Twelve.”

### TUTOR: “What is six times six?”

### STUDENT: “Oh, that’s a big number.”

(The student could multiply with very small numbers, but got nervous when larger numbers were asked.)

### TUTOR: “Six times six would be adding six to itself six times. Can

### you do this addition and tell me the sum?”

### STUDENT: (Pause) “Oh! I don’t like adding either.”

(The tutor then demonstrated the process of addition as “counting together.”)

### TUTOR: “Adding is counting numbers together. Are you comfortable

### with counting?”

### STUDENT: “Yes, I can count. One, two, three…,”

### TUTOR: (Stops her at the count of twenty) “Very good. Now count

### for me starting from eight hundred ninety.”

### STUDENT: (Taken aback) “Oh! That is a big number… (thinking) eight

### hundred ninety-one, eight hundred ninety-two… (and so on) eight hundred ninety-nine (long pause) two hundred, two hundred one…”

The student did not know what number followed eight hundred ninety-nine. By this time it was evident that the student was shaky in her understanding of the numbering system itself. She was not aware of the repeating pattern of hundreds in counting. The troubleshooting was ended at this point. The mother was given a program to establish an understanding of the numbering system first before working with her daughter on multiplication.

Evidently, the understanding of MULTIPLICATION depends on an understanding of ADDITION, which in turn depends on an understanding of COUNTING and the NUMBERING SYSTEM. If a person has simply memorized the sequence of first few hundred numbers, and has no understanding of the patterns of tens, hundreds, thousands, etc., he or she will have difficulty not only in counting with large numbers, but also with addition and multiplication. This principle of earlier “blanks” applies not just to mathematics but to any subject one is having difficulty with.

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