**The Feeling of Knowing When One Has Solved a Problem**

*when*a problem has indeed been solved. In working with undergraduates who have difficulty in solving problems, I have come to believe that a major aspect of the difficulty for some students is that they do not have an appropriate sense of when they have solved a problem. Two examples may serve to illustrate the difficulty. They are both taken from a Math Learning Skills course I taught which was designed to improve undergraduates' problem-solving abilities and to develop other skills needed in order to learn mathematics.

**A RATIO WORD PROBLEM**

The following ways of reasoning were offered by various students:I go to a certain place at 40 miles per hour and it takes me 20 minutes to get there. I return at 50 miles per hour. How long does the return trip take?

It takes 20 minutes to go 40 miles per hour. So you're going 10 miles every 5
minutes. (The student has blurred the distinction between miles and miles
per hour.) So to go 50 miles, it'll take an extra 5 minutes. So, it'll take 25 minutes. | |

It takes 20 minutes at 40 miles per hour, and 40 miles per hour 20 minutes; it
would take 10 minutes at 80 miles per hour, so 50 miles per hour 17½ minutes; it would take 15 minutes at 60 miles per hour, 60 miles per hour 15
minutes, which is half way between 40 and 80 miles per hour, 80 miles per
hour 10 minutes. So it would take 17½ minutes at 50 miles per hour which
is half way between 40 and 60 miles per hour. | |

It takes 20 minutes to go 40 miles per hour. (Again, the distinction between
miles and miles per hour is blurred.) So you're going 10 miles per hour for
every 5 minutes. But if you're going faster, like 50 miles per hour is faster
than 40 miles per hour, it'll have to take less time. So you subtract 5 minutes.
So it takes 15 minutes to get back. | |

You set up a formula, see: 20/40 = X/50. So it takes 25 minutes to get back. |

Each student was confident of his solution as he offered it, but registered no suprise when other students arrived at different solutions. I was able to lead the group to see the correct answer to the problem. My way of solving the problem did not, however, appear to seem more logical to the students than their own

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