Your own limits

Are you really almost there, or just starting out?

Tutoring, like all teaching, can be a frustrating business.  Take one student of our tutor-consultant: his grades in trigonometry have not been impressive.  So unimpressive, in fact, that he asked whether it was actually required by state law to pass the subject in order to graduate from High School.  (It isn’t.)  But as a student he is quick to grasp new ideas and can apply them correctly.  He has most of the subject in his head.  Unfortunately, he lacks a sort of overall structure, so that when presented with a test or homework problem he is likely to choose the wrong approach.  It’s like he has a ring of keys, and after trying one or two and finding they don’t fit the lock, he gets frustrated and loses patience.  He is convinced that he understands little or nothing, that he is in fact stupid as far as math goes, and has little chance of a decent grade.

In fact he is very close to mastering the whole subject.  With some patience, and a careful fitting together of pieces, everything is ready to fall into place.  There is the potential for that supremely satisfying bit of tutoring: when a small amount of help suddenly makes all of the student’s hard work come together.  Far from understanding nothing, the student understands almost everything.  His evaluation of his own competence is very wide of the mark.

Our astronomer points out that the opposite situation is probably as common, perhaps more so among research scientists.  In a book on popular astronomy published in 1878, four experts give their views on the conditions and physical processes occuring on the Sun.  They are wrong on almost all points.  It is true that key parts of Physics for explaining the Sun were unknown in 1878.  The power souce, nuclear fusion, wasn’t worked out even in principle for another half-century; the behavior of plasmas in strong magnetic fields, more difficult to calculate in detail, had to wait even longer.  One cannot fault these four scientists for coming up with the wrong answers.  But they express their views with great confidence, sure that they are mostly right and only need a bit more work to finish out the picture.

When you do have a good grasp of the subject, it’s pretty clear: you can deal with the homework or test problems correctly, subject to only minor mistakes.  (It is possible that these problems don’t really test your understanding.  That comes down to the teacher who organizes the course.  We may write more about this later.)  If you have no grasp at all, it’s also clear: you can gaze at the problem for long periods of time with no idea what is being asked or how to go about answering it.  The difficulty comes in between.  If you think your mastery of the subject is much lower than it is, your morale will be poor and you may give up.  If you overestimate your understanding, you may find yourself spectacularly wrong.

So how do you tell?  We’re still working on that.