On Avoiding Omissions in Mathematics
From a personal perspective, those who engage in mathematics often prefer brevity, sometimes omitting steps in transformations or providing proofs of theorems without explaining the process of discovering the theorems.
There is a term “amakudari” (天下り) used in Japanese that implies a method or solution is given in a sudden or unexpected manner, with no explanation of why it works, yet it somehow succeeds. This term describes situations where solutions are presented in a format that seems to work without clear justification.
I personally dislike “amakudari” solutions. Such approaches may be useful for solving a specific problem and memorizing the method can help with similar problems. However, relying on something akin to “divine revelation” to solve problems can leave one at a loss when that insight is not available. Fundamentally, one should rely on logical reasoning.
Of course, there are cases where working backward from the answer rather than transforming equations from the problem itself might be clearer. For example, expanding an equation involves mechanical operations, while factoring requires a certain conceptual approach. Differentiation is a mechanical process, but integration requires a conceptual understanding. Sometimes it is easier to teach the end result and have students work backward. Familiarity with these patterns can certainly improve one’s ability to solve problems.
However, mastering calculation techniques does not equate to doing mathematics. Being able to quickly solve known problems does not mean one can handle unknown problems. To solve unknown problems, one needs to understand at least “why one sets things up a certain way.” Repeatedly working through the reasoning can lead to understanding, but the initial resistance from beginners can be significant, and creating such friction unnecessarily is not advisable.
Of course, there are constraints of space. Writing out all transformations or drawing detailed graphs takes up space and can be laborious. Many mathematics textbooks also use “proof omitted” or “solution omitted” due to these constraints. Yet, from the perspective of helping beginners understand, this is a poor practice. While some might argue that filling in the gaps develops mathematical skills, it is not always feasible to do so perfectly.
Although some value the struggle as a virtue, I believe unnecessarily burdening young learners with difficulties has a negative impact on societal development (and, in this case, the development of mathematics). If a young learner, held back by the vast gaps, mistakenly believes they lack the intelligence for mathematics and gives up, that is a significant loss.
This isn’t to say that explanations need to be verbose or overly detailed. Each transformation should be explained concisely, with additional brief explanations for each step. Utilizing graphs and diagrams to aid visual understanding can greatly increase the number of people who grasp mathematical concepts. Mathematics should be understandable for everyone, even if it is complex (as long as it is mathematics). The difficulty in mathematics partly arises from the poor habit of omission by mathematicians and authors.
The origin of the habit of omission is unclear, but Gauss, for instance, often submitted only the finished product of his work and did not elaborate on the process. Various reasons have been suggested, one being that discovering beautiful truths took precedence over others’ understanding. In this sense, mathematicians are akin to artists, and expecting them to adopt an educational perspective might seem unsophisticated. Nevertheless, explaining truths from an educational viewpoint can attract like-minded individuals and bring forth new masterpieces.
Since there are no physical constraints in electronic media, I intend to write this article with as few omissions as possible.