In the two previous articles reposted on Scientific Space, matrix67 and Fan Xiang both expressed their dissatisfaction with most current (mathematics & physics) textbooks and offered some suggestions for writing them. Today, BoJone will also grumble a bit and share some thoughts on textbooks.
First, I must clarify that currently BoJone is only a high school sophomore, or rather, a high school student who loves mathematics and physics. Therefore, the views described in this article are merely personal opinions and likely contain many immature perspectives. Regardless, I present them here and welcome discussion.
BoJone believes that humans have a tendency to pursue benefits. If something is "good" for us or brings us convenience, we are very happy to possess or learn it. Mathematics and physics theories should be the same. When textbook writers want to introduce a new concept or present a new theory or method, the first thing they should do is not to rigorously define, derive, and prove it before finally applying it. On the contrary, they should write extensively about what "benefits" the introduction of the new concept or method brings. Only after understanding its utility will readers have a clear purpose and enough motivation for further study. This step is crucial for learning abstract theories; otherwise, tedious and dry reasoning and proof processes will kill the confidence of the vast majority of people. Even if they "finally" understand its use later, their interest will have significantly diminished.
Speaking of this, I must criticize a repulsive practice in the People’s Education Press (PEP) elective mathematics textbooks. In Elective 2-2, it introduces complex numbers, but only briefly explains the operations of addition, subtraction, multiplication, and division, along with definitions like the modulus, and then stops. Regarding the essence of complex numbers—for example, that complex multiplication represents coordinate rotation—it mentions nothing at all. What is the meaning of such "complex numbers"? A classmate once asked me: "What is the use of learning complex numbers?" I could only answer: "For now, the only use of complex numbers is to increase our burden in the Gaokao (College Entrance Examination)."
At the same time, BoJone has found that textbook writers have a very vague understanding of history and rely on too many "assumptions." The most obvious example is when discussing "why complex numbers were introduced." The textbook’s answer is: to handle the case where x^2+1=0 has no solution, the imaginary unit i was introduced. But is that really the case? Every high school student knows that when solving a quadratic equation, as long as the discriminant is negative, we loudly declare "this equation has no solution." Why would there be a need to introduce complex numbers? In fact, complex numbers were introduced "out of necessity" to handle the problem of solving cubic equations. This is because mathematicians discovered that when a cubic equation has three real roots, the calculation involves the square root of negative numbers. Thus, they reluctantly invented "imaginary numbers" (Imaginary numbers). As for their utility, that came later. But why don’t textbooks explain this clearly? Obviously, introducing complex numbers to students only through x^2+1=0 makes them very confused. They will wonder: "Why did you say there were no solutions before, but now you say there are two? Why do this?" and so on.
Regarding history, BoJone has deep feelings. It could be said that BoJone’s mathematical journey began with a book called Selected Lectures on the History of Mathematics. Our learning is a process of starting from nothing, and the accumulation of mathematical knowledge throughout history is also a process of starting from nothing. Both proceed step by step; therefore, history serves as a great reference. One might even say we should learn according to historical development. Whichever mathematical knowledge appeared first in history should be learned first. Because ancient mathematical knowledge focused more on practical applications, we need this knowledge more than complex and tedious theories. Furthermore, we should even repeat the detours and mistakes mathematicians made while researching problems! Because only by personally doing and failing can we gain deep understanding and potentially produce more creative research results. Additionally, studying history can increase our passion for research. When I see those exciting mathematical stories, I always get exceptionally excited and feel the beauty of mathematics deeply. For example, stories like the "Cubic Equation Duel" and the "Brachistochrone Competition" always make me read them over and over again. I never feel I’ve had enough, and sometimes I imagine myself as the protagonist, experiencing the academic atmosphere.
As for another problem with textbooks, it is the lack of "intuitiveness." Many concise and beautiful derivations have appeared in history. Although these derivations may lack rigor, they are manifestations of mathematicians’ creative thinking. We should savor them carefully rather than immediately thinking about how to prove them rigorously. Some wonderful examples, such as Euler extending the relationship between roots and coefficients of finite equations to infinity to conclude: \frac{\pi^2}{6}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... and Bernoulli using an analogy between Fermat’s principle of geometric optics and mechanics to find the equation of the brachistochrone, should make us applaud in admiration. As the first step of learning, we should first understand these "non-rigorous" contents and then think about how to make them rigorous. Textbook writing should include more of this content and place it at the beginning of each chapter rather than simply as an "appendix." Because these derivations are highly creative, in BoJone’s view, "creativity" is slightly more important than "rigor." Imagine when we open any modern mathematics textbook on any subject; if we are faced with abstract symbolic reasoning completely disconnected from our sensory experience of the actual world, how pessimistic that would be! It is difficult to learn creative thinking from such textbooks. In this regard, I greatly admire books like What is Mathematics? and Visual Complex Analysis. The authors sometimes intentionally discard rigorous proof processes to introduce a large amount of creative reasoning, which is particularly fascinating.
Textbooks should teach us not just "how to calculate," but also "why we calculate this way." Some complex problems become much simpler after a transformation, and we cannot help but applaud. However, most textbooks do not mention a single word about "from what perspective one thought to obtain this transformation." For example, in The Three-Body Problem by Wang Jiahe, he introduces the use of elliptic coordinate transformations to solve the "two fixed centers" problem, but he completely fails to mention how to think to arrive at this transformation, leaving me feeling very lost while reading. Therefore, textbooks should not just simply list problems and answers but should teach us how to think. Another example: in the vast majority of theoretical mechanics textbooks, when discussing the Principle of Least Action, they mention the action for Newtonian mechanics, special relativity, and so on, but few people mention "how to find the action" or "what characteristics the action possesses."
That’s all for now... ^_^
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