[This is the third essay on Study from 1996.]

“Blanks” in understanding cannot be filled properly unless one starts out very simply and then proceeds on an easy gradient of increasing complexity.  For example, when tutoring on addition, one should start out with single-digit numbers before working with double-digit numbers and carry overs.

In a Math Club meeting, a second grader was having difficulty writing large numbers in spite of repeated attempts by the parent to assist her.  A troubleshooting session went something like this:

TUTOR:           “Is it ok if I ask you to write some numbers for me?”

STUDENT:      “Yes.”

TUTOR:           “Alright.  Can you write six thousand, seven hundred


STUDENT:      “Umm…”

TUTOR:          “That’s ok.  See if you can write seven hundred eighty-


(The student thinks for a moment and writes “700 83”.  The tutor noticed that she could write eighty-three correctly.)

TUTOR:           “Ok.  Can you write eighty-three for me?”

(The student smiles and writes “83”.)

TUTOR:           “Excellent.  Can you write one hundred?”

(The student writes “100” correctly.)

TUTOR:           “Very good.  Now, can you write one hundred one?”

(The student writes “101” correctly.  The tutor then asked the student to write “one hundred nine” and “one hundred ten”.  The student wrote them correctly.)

TUTOR:           “Excellent.  Can you write one hundred eighty-three?”

(The student pauses then writes “183” correctly.)

TUTOR:           “That is correct.  Now write seven hundred eighty-three

  for me?”

(The student feeling more confident writes “783”.)

The troubleshooting session was ended at this point.  The parent then continued in this manner with the student writing larger numbers successfully.

This demonstration illustrates the necessity that a student’s understanding must be established at each step on a gradient for learning to occur.  This can be done only by letting the student assimilate the data by himself or herself.  The tutor must not think for the student.


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