Onmatu- General Math Learning Article
As I said in the previous article the student must examine carefully, several times and under various aspects the main parts of the problem. If the problem is related to a drawing, then this should have also the unknown and data on it. If we need to choose different names for these objects, the student must enter corresponding notations. Giving adequate attention to the election of symbols, the student has the obligation to examine the objects for which he needs to choose the symbols. There is another question which can be also helpful in this preparation stage, but with the condition to not expect a final answer, but a provisional one which the student intuits or guesses: can the condition be satisfied?
8. Example. Let's illustrate some of the matters set out so far. We will choose the following simple problem: Let's find the diagonal of a cuboid for which we know the length, the width and the height.
To discuss usefully this problem, the students must be familiar with the Pythagorean Theorem and with some of its applications in the plane geometry, but can have very little systematic knowledge in the field of space geometry. Here, the teacher can use the knowledge of the space relations, which the student has it from everyday life. The teacher ca make the problem interesting, embodying it. The classroom is a cuboid, of which dimensions can be measured or evaluated with approximation; the students must find, "measure indirectly" the diagonal of the classroom. The teacher shows which are the length, the width and the height of the classroom, points with a gesture the diagonal and gives life to the figure drawn on the board by repeated references to classroom.
The discussion between the teacher and the student can begin this way:
Teacher: "What is the unknown?"
Student: "The length of the diagonal of the cuboid."
Teacher: "What are the data from the problem?"
Student: "The length, the width and the height of the cuboid."
Teacher: "Choose the according notations. What letter you need to put for the unknown?"
Teacher: "What letters you need to choose for the length, the width and the height?"
Student: "a, b, c"
Teacher: "What is the condition that bounds between them a, b, c and x?"
Student: "x is the diagonal of the cuboid which has the length, the width and the height equal with a, b and c respectively"
Teacher: "Is the problem having a sense? I want to say: Is the condition sufficient to determine the unknown of the problem?"
Student:"Yes, it is sufficient. We know the cuboid because we know a, b and c. The cuboid being determined, the diagonal is determined."
Now the teacher who wishes to develop the ability of his students to solve problems must instill in their minds some interest for problems and give them all the possibilities to imitate and exercise. If the teacher wants to develop at his students the intellectual appropriate operations for the questions and recommendations from our list, he will put the students the same questions and make these recommendations whenever the opportunity presents itself naturally. More, when the professor solves a problem in front of the class, he must direct his ideas a little and put himself the same questions he uses to ask the students. Guided this way, the student will discover eventually the correct use of these questions and recommendations, thereby gaining something more valuable then just knowing some particular mathematical facts. Thank you.