A Gap in Mathematics Education

The process of creation of mathematics has the following hierarchically dependent components:

• Coming up with a concept.
• Coming up with a question dealing with a relationship between concepts (this includes formulating a hypothesis, as well as finding an example or a counterexample of a concept/phenomenon).
• Answering a question dealing with a relationship between concepts (this includes proving theorems as well as solving problems without being given the recipe for solution).
• Applying the answer to a question dealing with a relationship between concepts to answer another such question (this includes solving problems by applying a given recipe for solution).

Modern mathematics education (both at the school and at the university levels) focuses mainly on the last two points. What is regarded as a low quality mathematics education would focus only on the last point. For a more whole mathematics education, the first two points must receive as much attention as the last two points do. 

It is not difficult to implement the first two points in the practice of mathematics teaching. Here is an example of the structure of a class that focuses on the second and the third points:

1. The teacher proposes one or two concepts that the pupils are familiar with (perhaps, by taking suggestions from the class). 
2. The teacher then asks the pupils to explain the concepts, helping the pupils in the explanation, when necessary. 
3. Then, the teachers asks the pupils to think of a question that would combine the named concepts. The teacher helps in this process.
4. After this, the teacher and the pupils engage in answering the question together. 
5. If the question is too hard to answer, it should be concretized to a simpler question. If the question is too easy to answer, it should be abstracted to a more difficult question.

Concepts arise in mathematics as a necessity to help one express a general phenomenon. Incorporation of the first point in a classroom can be achieved by explaining this necessity for the concepts that the pupils are already familiar with, or by taking pupils on a journey that would help them identify such a necessity and will result in (re)discovering a mathematical concept. Teaching concepts by first showing examples and then asking the pupil to develop a concept that fits those examples is another, perhaps simpler, way. The activities which ask a pupil to identify a pattern in a sequence of numbers or figures is in some sense of this type. However, these activities are sold as activities that fall under the third point, as the pupil is being convinced that the question must have one definite answer.