This article was converted by Jianyue SimpRead; the original source is www.zhihu.com
Answer One
Online praise for foreign textbooks generally comes from university students or graduates who favor overseas university-level textbooks. This is because, at the university level, time spent studying—especially on repetitive material—is significantly lower than in primary and secondary school; thus, diligence in problem-solving can no longer mask flaws inherent in textbooks. In fact, primary and secondary school textbooks also suffer from numerous issues.
Let’s begin with a mistranslation in mathematics: rational numbers and irrational numbers. Doesn’t it sound odd? What does “rational” or “irrational” even mean when applied to numbers? Do numbers possess “reasonableness” or “unreasonableness”?
In English, rational numbers are called rational, and irrational numbers irrational. While rational indeed carries the meaning of “reasonable” or “logical,” in mathematics it derives from the Latin root ratio, meaning “ratio” or “proportion.” Thus, rational number literally means “number expressible as a ratio (of two integers),” while irrational number means “number not expressible as such a ratio.” A number is rational if it can be written as the ratio of two integers; otherwise, it is irrational.
This translation error has serious consequences: many Chinese people—even university graduates—forget what “rational” and “irrational” actually signify if they haven’t encountered these terms for some time.
Next, consider the term equation. Its English equivalent is equation, meaning simply “equality.” A chemical equation is thus a chemical equality, expressing equivalence between substances before and after a reaction. Likewise, a physical equation expresses equality between physical quantities before and after motion—for instance, conservation of momentum or conservation of mechanical energy. The Chinese term fangcheng (equation) gives no intuitive clue about its meaning—“equality”—leading many Chinese learners to lack the higher-order conceptual understanding of “equality” altogether.
Below is a Grade 7 mathematics textbook from California. When teaching word problems involving equations, it first requires students to write out the equality explicitly. All subsequent steps—addition, subtraction, multiplication, division—are performed while preserving equality, rather than using China’s common “transposition” method. For example, in Step 3 shown below, both sides are simultaneously subtracted by 3, maintaining equality—not “moving” the 3 to the right-hand side and changing its sign to −3. To compensate, Chinese educators have even invented mnemonic rhymes for transposition—a technique far less mathematically rigorous than simultaneous operations on both sides.
In contrast, Chinese instruction on equation-based word problems begins by defining unknowns. Typically, students immediately assign an unknown variable based on what the question asks; expressing relationships via equalities is treated merely as an incidental step. Although Chinese students practice many problems and become highly proficient at solving equations, their grasp of the fundamental concept of “equality” remains vague. Consequently, when tackling physics problems—which frequently require setting up equations (equalities)—they often resort to mechanical imitation rather than consciously, systematically, and insightfully identifying equalities among physical quantities under the guiding principle of “equality.” Indeed, one of the core difficulties in high-school physics lies precisely in discovering such equalities—not in selecting appropriate unknowns.
Even more puzzling is the inconsistent terminology used for variables or unknowns. In function theory, we speak of independent and dependent variables, yet in equations, the same objects suddenly become “yuan” (literally “elements” or “units”). For example, solving a system of two equations is taught as “eliminating one yuan” to solve for the other, then substituting back to find the first. Does the literal meaning of yuan clarify this procedure? Not at all. Only by interpreting them as variables or unknowns does the logic become clear: an English textbook calls this an equation in two unknowns. With two unknowns, simultaneous solution is impossible; hence, one must first eliminate one unknown, reducing the system to a single-equation problem with only one unknown—then solve and substitute. A typical Chinese middle-school student may mechanically solve such systems with great fluency, yet remain unaware that “elimination” refers to removing or reducing unknowns—or even fail to notice that unknowns are being eliminated at all. Later, when confronted with basic inequality problems in high school, they stall completely.
Thus, online tutoring teachers commonly compile exhaustive lists of “standard methods” for basic inequality problems, urging students to memorize them verbatim—an inefficient, labor-intensive, and ultimately unhelpful approach. In essence, all such techniques aim to eliminate variables or unknowns, since maximum and minimum values contain no variables or unknowns.
Many of the above approaches rely on multiplying numerator and denominator to cancel unknowns. For instance, Method 2—called the “matching method”—essentially ensures the denominator contains (x − 2), then arranges the numerator also to contain (x − 2) so that multiplication cancels x. Method 3—named the “substitution of 1”—uses the denominator twice to multiply the numerator (2a + 3b), thereby eliminating both variables a and b. Method 5—the “common division method”—starts from ab = (a + 2b)/3, whose minimum value is sought. Since the numerator is (a + 2b), dividing both sides by ab yields 1/b + 2/a, enabling cancellation of a and b upon multiplication. Is it necessary—or even beneficial—to categorize these techniques so finely and teach them as distinct methods? Such excessive segmentation overwhelms students, making retention nearly impossible. Most classroom instructors rely on intuitive, ad-hoc “magic tricks” born of experience—exactly like the examples above—without fully grasping the underlying principles behind their algebraic manipulations.
Another problematic term is differential. This translation fails to convey the concept’s most essential mathematical meaning. In English, differential stems from difference, inherently implying change. A differential quantifies rate of change—how fast something changes. Yet the Chinese term weifen (“micro-differentiation”) conveys only “division” or “fine subdivision,” omitting any sense of change—thus straying far from its true mathematical purpose. As a result, many learners apply differentials mechanically, imitating procedures without deep conceptual integration.
Similarly, partial differential poses comprehension challenges for beginners. Can the term itself yield immediate clarity? Unlikely—but its English counterpart partial differential (“partial change”) directly reveals its function. Partial primarily means “part” or “portion”; it’s unclear why translators chose the connotation of “bias” or “prejudice” instead.
The term tangent line—translated in Chinese as qie xian (“cutting line”)—is another case in point. Its English name tangent line reflects its central mathematical role: the slope of a tangent line equals the tangent (trigonometric function) of the angle it makes with the horizontal axis. Removing “zheng” (“true” or “positive”) from the Chinese rendering zheng qie xian obscures this connection entirely, depriving students of valuable linguistic scaffolding for conceptual understanding.
In linear algebra, the term pivot denotes the first nonzero entry in each row of a matrix reduced to row-echelon form. In English, pivot literally means “axis of rotation”: elementary row operations effectively rotate rows around the pivot point. Translating it as zhu yuan (“main element”) discards this geometric intuition completely.
Likewise, the determinant—determinant in English—derives from determine, meaning “to ascertain” or “to decide.” Its actual function is precisely to determine whether a matrix is invertible. The Chinese translation hang lie shi (“row-column formula”) utterly erases this functional meaning, leaving many undergraduates puzzled about the determinant’s purpose.
In artificial intelligence, the term regularization is rendered as zheng ze hua (“regularization”), offering no hint of its mathematical role. Its English root regulate means “to adjust” or “to calibrate.” Mathematically, regularization achieves exactly that: adding a small penalty term to the loss function to tune model parameters.
Physics introduces further confusion with velocity and speed: one is a vector, the other a scalar. In everyday Chinese usage, su du (“speed”) corresponds to the physics concept of speed (scalar), yet beginning physics students frequently conflate the two. Even after learning linear velocity and angular velocity, Chinese learners must pause to recall whether each term refers to a vector or scalar—and over time, forget entirely. While velocity, linear velocity, and angular velocity retain clear vectorial meaning in English, the term uniform speed in “uniform linear motion” is likewise vectorial—yet paradoxically, uniform speed in “uniform circular motion” is scalar! In English, speed is always scalar; velocity, linear velocity, and angular velocity are consistently vectorial.
The physics term conservative force originates from conserve, which indeed carries the sense of “conservatism,” but this translation unnecessarily burdens students with conceptual confusion. In Chinese, bao shou (“conservative”) implies “old-fashioned” or “reactionary”—so how does that relate to physics? In reality, conserve also means “to preserve” or “to retain.” A conservative force preserves mechanical energy during motion—energy isn’t lost. Crucially, the English Law of Conservation of Mechanical Energy uses the same root conserve—whereas Chinese splits it into bao shou li (“conservative force”) and shou heng (“conservation”), creating artificial lexical disconnection.
The term frictional force itself is unproblematic, but its standard explanation creates widespread confusion. Friction is typically defined as follows: “When an object moves—or tends to move—along the tangent direction of contact with another object, a force arises at the interface resisting their relative motion; this force is called friction.” This phrasing misleadingly suggests friction arises from motion, and some careless texts even claim outright that friction is caused by relative motion—contradicting the fundamental definition of force. Students may not even realize that, in their minds, “force” now carries two conflicting definitions: (1) something that changes motion state, and (2) something produced by motion—the latter being incompatible with the rest of mechanics and thus breeding persistent confusion.
For problem-solving, the greatest difficulty lies in determining “tendency toward relative motion”—key to establishing friction’s direction. Students repeatedly second-guess this tendency, frequently erring. This difficulty stems entirely from the inappropriate definition used in Chinese textbooks. The English definition—“a force opposing relative motion between contacting surfaces”—contains no mention of “tendency.” Why? Because English defines friction as resistance to actual or impending relative motion, whereas the Chinese phrasing—by implying friction arises from motion—forces inclusion of “tendency” to cover cases where no motion occurs.
Consider the following high-school physics problem. Determining “tendency toward relative motion” on an inclined plane is ambiguous: the object might tend to slide down—or up. Relying solely on “tendency” leaves students bewildered. The author proposes using the assumption method—which essentially abandons “tendency” entirely and reverts to standard force analysis grounded in the proper definition of force.
Friction’s definitional issue epitomizes broader flaws in Chinese textbooks: an ill-chosen definition leads students to fail at problem-solving; tutors devise workarounds bypassing the definition—but nobody questions why such workarounds are needed. Less solid learners end up thoroughly confused, unable to reconcile when the textbook definition works versus when it doesn’t.
Physics problems also love playing with language. The word instantaneous (shun jian) is perhaps the most abused term in high-school physics exams. It rarely appears in textbooks yet features heavily in tests—not in the calculus sense of limits, but embedded in conventional contextual assumptions. For example, “the elastic force in a massless ideal spring cannot change instantaneously,” whereas forces in massless rods or strings can. Problems seldom state explicit constraints on instantaneous force changes; instead, they merely drop the word shun jian, forcing students through unnecessary mental detours. The result? An ideal massless spring suddenly behaves non-ideally—failing to deform promptly.
Some argue vocabulary bears little relation to mathematical understanding—that only conceptual mastery matters, regardless of linguistic labels. Hilbert held similar views: in his axiomatic geometry, he deliberately left “point,” “line,” and “plane” undefined, constructing logical relations solely via five groups of axioms. He asserted these primitives could be replaced by any concrete objects—“tables,” “chairs,” or “beer mugs.” But imagine actually teaching geometry using “table,” “chair,” and “beer mug”—most students would be instantly lost. Point to a dot and call it a “table”—confusion is inevitable.
Language profoundly shapes mathematical understanding. As noted in The Number Sense:
“Surprisingly, language differences cause U.S. children to lag behind their Chinese peers by up to one full year. At age four, Chinese children can typically count to 40; American children struggle to reach 15 at the same age—and require a full year to catch up and count to 40 or 50. This gap doesn’t persist for smaller numbers: both groups perform equally well up to 12. Trouble arises with ‘13’ and ‘14’—numbers whose English names violate base-10 regularity, whereas Chinese names follow it perfectly. Chinese spoken numerals align precisely with Arabic notation, easing mastery of base-10 place-value concepts. When asked to construct ’25’ using unit cubes (value 1) and ten-rods (value 10), Chinese children effortlessly select two ten-rods and five cubes—demonstrating grasp of base-10 structure. Their American peers mostly fail to use ten-rods efficiently, painstakingly counting out 25 individual cubes. Worse, given a twenty-rod, they usually choose it over two ten-rods—focusing superficially on the word twenty-five, while Chinese children already internalize deeper base-10 architecture. Base-10 is intuitively obvious across Asian languages but notoriously difficult for Western children.”
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Commenters note induction formulas—another classic mistranslation.
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International Standard Term: In global mathematics, the standard English term is reduction formula, meaning “formula for reduction”—accurately reflecting its function: reducing trigonometric functions of complex angles to those of fundamental angles.
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Origin of Chinese Translation: The Chinese term you dao gong shi (“induction formula”) originated from the Russian Formuly privedeniya (“conversion formulas”). When 20th-century Chinese textbooks adopted Soviet materials, this was mistranslated as you dao (“induction”)—and the error persists today.
Answer Two
Honestly, American textbooks are truly exceptional.
How exceptional? If you read them carefully, you won’t even need a teacher.
Select a textbook from a major publisher (HMH, McGraw-Hill, Pearson), and study chapter-by-chapter—you’ll become unstoppable.
Too many commenters ask where to buy these textbooks. Here’s a useful site—I believe it covers most undergraduate-level content: https://open.umn.edu/opentextbooks/subjects#
Note: This site offers free open-access textbooks, but they’re not necessarily the latest or most popular commercial editions.
— Update —
Given this post’s popularity, I’d feel remiss not sharing practical resources. Many top U.S. universities offer free online courses:
Harvard Online Courses: https://pll.harvard.edu/catalog/free
Stanford Online Courses: https://online.stanford.edu/free-courses
Carnegie Mellon OLI: https://oli.cmu.edu/independent-learner-courses/
But my personal favorite remains Khan Academy—especially for humanities. You can listen comfortably while driving, even without watching the screen: https://zh.khanacademy.org
Wishing all students diligent effort and rewarding growth.# Response Three
I ranked among the top 1,000 students in Sichuan Province’s college entrance examination (Gaokao), scoring over 130 in mathematics. While browsing a street-side stall, I stumbled upon a set of the Self-Study Series for Mathematics, Physics, and Chemistry, originally published in the 1980s—absolutely stunning! After flipping through the entire set, I deeply regretted not acquiring it earlier; I might have even scored a few extra points on the exam.
This series is truly ideal for self-study—especially its algebra volume. Upon completing it, you’ll experience realizations like:
“Ah, so this unit is actually about this!”
“The most crucial concept is this!”
“So this is how important concepts are tested!”
Older textbooks feared you wouldn’t understand enough; today’s textbooks fear you’ll understand too much!
It’s truly unfortunate—such outstanding (and fundamentally rigorous) high-school textbooks are no longer available domestically.
Below is the digital resource—feel free to check it out if interested.