All too frequently, problem solutions are attempted by manipulating numbers given in the problem statement. The use of ideas then is overlooked. By responding directly to what is ASKED FOR
The following problem ASKS FOR the age of the shepherd. The logical structure is generated by using
SureMath - the math software for 21st century problem
Skills in manipulating numbers are developed at a quite early age. The transition to using these skills appropriately, such as in problem solving, is done in a quite haphazard manner. The student then does not have an opportunity to get past the rote number manipulation stage and will then resort to rote number manipulation to solve problems.
It has been observed that a significant fraction of the children who have been tested with this problem produce a numerical result. This is as it should be. What is missing in such students' solutions is appreciation of the rationale used to obtain a numerical result.
The cognitively capable children who produced the numerical result had a basis for the choice every bit as sound as the one used above. However, they may not consciously realize that any sensible answer is correct and that a sensible answer can be obtained by any sensible method.
Learning to explore missing-information problems systematically, in a meaningful way, adds significantly to the students' cognitive abilities.
After all, this is the process one uses in daily affairs to supply a reasonable value for something that is not known in a particular situation.
Some may find "off-the-wall" assumption of the age of the shepherd being 1/2 the sum of the number of goats and sheep annoying. This is of course exactly the point. Any choice can be made to supply the missing information that produces a resonable answer. One can replace the assumption made here with any other statement which serves this purpose. This is used to generate new problems.
In any case the fact that some students obtain an numerical result for the problem as stated with missing information is a positive indication of thinking and is commendable. It shows an intuitive recogonition of mathematical statement that a system of linear equations containing more unknowns than equations usually has infinitely many solutions. The "off-the-wall" assumption made above simply makes use of this principle.
Additional information about problems of this type can be seen in
Revisiting Mathematics Education: Chine Lectures by Hans Freudenthal, Dordrecth, Kluwer, 1991.