The journal is not officially founded by 58, and the content only represents the author’s personal views.
Author’s Profile: Wu Di, a graduate of Qingdao No. 58 High School, class of 2024, scored 662 in the college entrance examination (137 in mathematics) and was admitted to Xi’an Jiaotong University, currently studying in the university’s elite computer experiment class.
Abstract: This article is based on my understanding of the mathematics subject during high school and an investigation into the learning conditions of students with varying levels of mathematics proficiency. Using some questions from the 2024 New College Entrance Examination I卷 as examples, I will systematically introduce the innovative method of learning mathematics through “natural language processing.” Please note that everyone’s learning methods are different, so the methods here are not universal, especially not suitable for students whose current math level has reached above 130. If your math level is between 80 and 120, you can prioritize my methods.
Mathematics is a “Language”
Some say “mathematics is a verb,” but I don’t think that’s entirely accurate. This statement overly emphasizes the role of practicing problems. Theoretically, practicing problems does have a high ceiling; for example, I can score full marks on simple and medium-level questions through practice, and my final score should be around 137. However, the lower limit of practicing problems is also very low. Considering that the academic burden in high school is very heavy, just practicing problems can lead to physical and mental exhaustion. If there are no short-term results, it can also produce fear and frustration (don’t PUA yourself by thinking these emotions arise because of your “lack of endurance”; in fact, these emotions guide you to change your learning approach!). Students who like to sharpen their pencils often treat mathematics as an engineering subject to learn.Many people study mathematics without knowing what mathematics really is. Many people study mathematics, stacking problem sets, error notebooks, and notes high, but in reality, they have never had any organized class notes or notebooks, which does not hinder their mathematics from scoring high. Those who like to take notes often treat mathematics as a humanities subject to learn.
However, mathematics is not simply a “subject,” but a “language”. Think about how we learned set theory when we first entered high school; we built the “foundation” of this language. In the first chapter, the fifth section, didn’t it directly write the word “terminology” in the title? Similarly, every mathematical concept we learn in the following sections constitutes a dictionary that “maps” mathematical language to natural language. Let’s look at some examples:
-
Derivatives: The speed of the function’s “rise and fall” -
Variance: How “spread out” the data is -
Always true and sometimes true: Always true is “strict (hard to achieve),” while sometimes true is “gentle (easy to achieve)”
Here, “rise and fall speed,” “spread,” “strict,” and “gentle” are all common natural language terms. Thus, complex definitions become very simple and clear. This also suggests that we can utilize these characteristics during problem-solving by translating problems into “natural language” to crack them.
Uses of “Natural Language”
While “natural language” is the “breakthrough” for many problems, it is not necessarily a “panacea” for everything. For example, some problems that test transformations assess “numerical sense,” which is another dimension of ability. The “natural language” mentioned here mainly applies to translating problem conditions and new definitions.
Translating Problem Conditions
Taking the 8th question from the 2024 New College Entrance Examination as an example:
Given that the domain of the function
is f ( x ) , R , and when f ( x ) > f ( x − 1 ) + f ( x − 2 ) , x < 3 , then which of the following conclusions is definitely correct? ( ) f ( x ) = x A.
f ( 10 ) > 100 B.
f ( 20 ) > 1000 C.
f ( 10 ) < 1000 D.
f ( 20 ) < 10000
For the second-order recurrence relation, we can easily think of some special sequences like the Fibonacci sequence. The Fibonacci sequence satisfies
grows “faster” than the Fibonacci sequence. f ( x )
Since
From the above example, it is clear that the so-called “translation into natural language” is to express the relationships (including equalities or inequalities, geometric properties, etc.) between numbers or figures in the problem in plain language, and then solve the problem according to the guidance of the “plain language.”
Translating New Definitions
Since the nine-province joint examination in 2023, new definitions have emerged, knocking down all the heroes who used to dominate the 130 score range to around 110, leading to widespread despair and complaints about “mathematics that supplements drug studies.” In fact, the new definition type of questions is the best place to apply the “natural language” method. For example, I created a new definition question for everyone using my final exam score of 83:
19. (17 points) Let
be a sequence, and { a n } be a real number. If for any given positive number A (no matter how small), there exists a positive integer ϵ such that when N , the inequality n > N holds, then the sequence ∣ a n − A ∣ < ϵ is said to converge to { a n } . A (1) Prove: The sequence
converges. { 1 /n } (2) Prove: A sufficient condition for an infinite increasing sequence to converge is that it has an upper bound (i.e., there exists a real number
such that each term of this sequence is less than or equal to M . In particular, the smallest M that satisfies this condition is called the “supremum.”). M (3) Let the sequence
satisfy { a n } , and for any positive integer a 1 = 1 , we have n . Determine whether the sequence a n + 1 = (1/2) a + 1 a n ) converges, and provide proof. { a n }
(Note: In the foreseeable future, the college entrance examination will not directly use higher mathematics knowledge for new definitions; here, higher mathematics knowledge is only used to illustrate the solution methods for this type of question.) Here, we can apply “natural language” to solve the problem. We can think of
can only survive in the narrow “gap” after reaching the “partition” { a n } . Moreover, the narrower the “gap,” the further back the “partition” is (i.e., the larger N is). N
Then we further process these concepts of “gap” and “partition.” The sequence
As
increases, the sequence n gets closer and closer to the real number { a n } and almost completely adheres to it. A
This way, we intuitively understand the new definition of the problem: “convergence” represents an asymptotic process. The first question is easy to solve. Now, how do we understand “supremum” in the second question? We still use natural language to understand this concept. We can continuously press down the “upper bound” and find that when we are very close to the largest term of
Reconstructing Natural Language
In particular, there is a type of question that goes against the grain, using very lengthy plain language to describe the background and conditions of the problem, which seems simple but is actually troublesome. This type of question is common in probability and statistics, and the solution method is, of course, to “simplify complexity” by reconstructing the complex language into language that conforms to mathematical thinking. For example, the 14th question from the 2024 New College Entrance Examination:
14. (5 points) Person A and Person B each have four cards, each marked with a number. Person A’s cards are marked with numbers
, 1 , 3 , 5 , and Person B’s cards are marked with numbers 7 , 2 , 4 , 6 . They compete in four rounds. In each round, both players randomly select a card from their own cards and compare the numbers on the selected cards. The player with the larger number scores 8 point, and the player with the smaller number scores 1 points, and then both discard the cards they selected in that round (the discarded cards cannot be used in subsequent rounds). What is the probability that Person A’s total score is no less than 0 after four rounds? 2
The best method for this question is, of course, brute force enumeration (but I had a classmate from Jiaotong University who enumerated to get
(Condition 1) Person A must have at least two cards that are larger than Person B’s.
Now, let’s look at poor Person A, who, due to bad luck, drew four small cards, can only use the strategy of “Tian Ji’s Horse Racing” to ensure that 7 must win first. More straightforwardly:
(Inference 1) When Person A draws 7, Person B cannot draw 8.
Next, we analyze the impact of drawing numbers 3 and 5, and similarly use the “Tian Ji’s Horse Racing” approach to think, and we can easily conclude:
(Inference 2) At least one of Person A’s 3 and 5 must win against Person B.
Next, consider the influence of the drawing order. Regardless of which round Person A draws 7, as long as they ensure that Person B does not draw 8 in the corresponding round, it will suffice. Similarly, for Person A’s 3 and 5, they just need to ensure that one of them wins. Therefore, we have:
(Condition 2) The order of drawing cards does not affect the outcome.
Thus, we have an enumeration idea:
First, consider all situations, with a total of
combinations; then enumerate the situations that satisfy the above conditions and inferences, totaling 24 situations, and obtain the answer: 12 . 1 /2
In the process of solving this problem, we can easily see that if we deconstruct the complex and compound conditions in the problem into easily judged conditions using natural language, we can simplify the data processing process, shorten the time, and increase the accuracy.
Usage of Natural Language
From the explanations of the above examples, it is clear that if we effectively use natural language as a tool to interpret problems, many difficult questions become straightforward. However, how do we come up with such elegant natural language to describe problems in the examination room?
Special Properties + Associations
For example, in the aforementioned 8th question, we notice the similar “second-order recurrence” property of the function and associate it with the Fibonacci sequence. Here,
Local Adjustments + Sentence-by-Sentence Translation
This method is often used in question 19. For example, in the sample I provided, we continuously adjust
Using Visual Language
Visual language is the most primitive and intuitive form of natural language. Here, visual language can be the Cartesian coordinate system, Venn diagrams of sets, or even conceptual diagrams. For example, for a problem like “three people competing,” if we can draw a tree diagram representing the win-lose relationships, it becomes much easier to list the events. Similarly, for abstract vector relationship questions, if we confine the vectors within a right triangle or square, the situation becomes much simpler. When applying this method, ensure that the situation is limited or follows a pattern, and that the drawings are clear and concise. At the same time, be cautious not to overly specialize the general situation in the drawing process.
Conclusion
Using natural language to solve problems is indeed a simple yet accurate innovative method. If this method and its underlying philosophy are applied to daily mathematics learning or review, it can deepen students’ understanding of mathematics while avoiding empty discussions of “high-level” outdated rhetoric and keeping it grounded.
Mathematics content continues to be updated, please stay tuned.
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