abstractmath.org
help with abstract math
Produced by Charles Wells. Home. Website Contents Website Index
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Posted 4 February
2009
DOING MATH: Self-monitoring
Pay attention to what you are
doing when you are
trying to do math. Ideally you should
install in your brain a kind of Watcher 
who watches what you do without being judgmental. Math Ed
people call this self-monitoring. Self-monitoring enables you to discover both functional
and dysfunctional behavior.
When you are doing math,
notice what works and what does not work
It takes work to succeed at self-monitoring in an area where you have not practiced it. I cannot tell you much about how to install the Watcher. All I can say is: Be aware of the usefulness of watching yourself work!
When you practice self-monitoring successfully while doing math
then in the long run you will learn more about doing math
than you will with any other practice suggested on abstractmath.org.
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That is because this website is a compendium of things known to help people struggling with abstract math. When you watch yourself doing math, you are learning about YOUR difficulties, not about other people’s difficulties.
Example
This is a baby example. Suppose you are told that your college has 6 times as many students as professors and that the total number of students and professors is 1400. How many students does your college have?
You start working. Let s be the number of students and p be the number of professors. So
Then ,
so
,
so
. So the college has 200 students.
That can’t be right! Let’s see,
and
. So plug
into
and you get
,
so
but
But now…
The Watcher says: You did the same thing twice and
got the same answer. Maybe it’s time to
try something else. Like
for example checking what you did!
Well, is right, that’s 6 students for every professor. No wait a minute. If there are 100 students there are 600
professors.
Rowrbazzle!
I should have written
…
Notes
¨ The example above is well-known to math ed people. The conceptual blunder is thinking of p and s as labels instead of as variables. The math ed people call it the “student-professor problem”. Seems to me every math misconception is a student-professor problem…
¨ Self-monitoring is a great help in other aspects of life besides math! Typically, people do it in certain types of activity and not in others (different for different people). Sometimes people avoid thinking about or grappling with problems that they have in certain areas of life, with the result that they don’t solve their problems in that area.