Lam [1], explained how mathematics is not only a technical subject but also a cultural one. As such, mathematical proofs and definitions, instead of simply numerical calculations, are essential for students when learning the subject. Hence, there must be a change in Hong Kong’s local teachers’ pedagogies. This author suggests three alternative way to teach mathematical philosophy through infinity. These alternatives are as follows: 1. Teach the concept of a limit in formalism through story telling, 2. Use geometry to intuitively learn infinity through constructivism, and 3. Implement schematic stages for proof by contradiction. Simultaneously, teachers should also be aware of the difficulties among students in understanding different abstract concepts. These challenges include the following: 1. Struggles with the concept of a limit, 2.Mistakes in intuitively computing infinity, and 3. Challenges in handling the method of proof by contradiction. Adopting these alternative approaches, can provide the necessary support to pupils trying to comprehend the above mentioned difficult mathematical ideas and ultimately transform students’ beliefs [2]. One can analyze these changed beliefs against the background of con-ceptual change. According to Davis [3], “this change implies conceiving of teaching as facili-tating, rather than managing learning and changing roles from the sage on the stage to a guide on the side”. As a result, Hong Kong’s academic results in mathematics should hopefully improve.

There are various schools of mathematical philosophy. However, none of them can be founded on mathematics alone. At the same time, there are two types of mathematical proof styles: Dialectic and algorithm mathematical proof. The relationship between proof and philosophy is to study philosophical problems with mathematical models. This type of proof is important to Hong Kong Secondary education. In addition, teachers should explain the connection between mathematics-based subjects, such as physics, so that lessons are more interesting rather than technical. Mathematics relates to nearly all other subjects, and as such has the role of a ‘public servant’ when it comes to serving them. One role of mathematics is to act as a ‘rational’ instrument for various subjects. This can be shown in many ancient human activities, such as Daoism and Liu Hiu, together with their symbolic representations. These examples are similar to Jewish culture; when discussing confidence, Abraham is often mentioned due to being the “Father of Confidence”. Thus, it may be said that mathematics is more than just a servant—it is also a cultural subject that has been recorded throughout history. To conclude, other than mathematical proof, Hong Kong teachers should also allow students to learn the cultural context behind various topics and subjects.