Inside the Math

Introduction

To understand how the Gaokao and SAT reflect different ideas of what it means to be "educated," it's important to look closely at the math itself. I will examine both how students are prepared and how they are tested.

In this section, I analyze a specific math topic: sequences. I chose this topic because it is an accessible topic that appears on both exams.
I first examine how it is taught through learning materials (a Chinese high school textbook and Khan Academy SAT prep), and then compare actual exam problems from the Gaokao and SAT.

This structure allows us to see:

How deeply each system teaches a topic during preparation.
How that depth (or lack of it) shows up in the college entrance exams.
What kinds of reasoning, habits, and mindsets each system ultimately values.

Sequences

Sequences are one of the core topics in high school math, usually introduced in Algebra 1 in the US.

A sequence is simply a list of numbers arranged in a certain order, usually following a specific pattern or rule. For example, adding 2 each time (2, 4, 6, 8, ...) is a simple sequence.

Let's take a look at how the topic of sequences are studied in a Chinese textbook. The following pages come from Chapter 2 of People's Education Press High School Mathematics A Edition, Elective 2-2. The textbook reflects a standard high school mathematics curriculum that students are expected to know for the Gaokao.

This sample problem focuses on the "Nine Rings" puzzle. The puzzle involves an interesting contraption made up of 9 rings and a bar. The goal of the game is to disentangle the rings from the bar in the fewest moves possible.

The details of the game aren't too important for us. What is interesting here is the thinking skills required to solve the problem. The textbooks walks readers through the critical thinking process necessary to derive the recurrence relation, a formula that expresses a term in the sequence as a function of its preceding terms. This turns the physical game into a formal, mathematical model.

Then, they show the algebraic steps to produce a closed formula. This requires knowledge of summing geometric sequences, as well as carefully considering cases where n is odd vs even.

This problem builds proof-oriented thinking by building up the solution from first principles, training students to be technical problem solvers. It also trains endurance; to be successful, students must be able to think critically through multiple steps.

Now let's take a look at some of the other practice problems provided in the textbook.

This type of problem is common in Chinese textbooks. This multi-part problem builds foundational understanding by asking guiding questions to the reader.

Given an arithmetic sequence, the problem asks the student to manipulate it in several ways, including removing a prefix of the sequence, removing specific indices, and removing based on divisibility.

It's not just about calculating a number. Students must reason about the definition of an arithmetic sequence, analyze the pattern of a sequence, and recognizing which properties of a sequence are preserved under different changes. Chinese students are expected to reason flexibly about what they're learning and think like real mathematicians.

Now let's pivot over to the SAT. One of the primary resources for SAT prep in the United States is Khan Academy, an online platform filled with instructional videos, guided practice problems, and quick strategies for tackling common math topics.

In the following examples we'll take a look at Khan Academy's preparation for sequences in its Algebra 1 course.

This is a particularly funny example. This "challenge problem" on Khan Academy's "Intro to Arithmetic Sequences" is a simple series of computations.

This problem requires only basic arithmetic and counting, as well as knowing what an "arithmetic sequence is" informally. There is no structural reasoning, no derivation, and no proofs necessary. This might be considered a trivial warm up question in China.

Let's take a look at another one that's a little more complex. Taking the form of a word problem, the student is required to categorize the sequence and complete a recursive formula.

It's still an easy problem that is based on recognizing that the word "doubles" maps directly to a geometric sequence. Finding the recursive formula is also easier than deriving the formula for the general term. This example highlights how SAT preparation emphasizes quick recognition of basic sequence types rather than deep modeling or formal proof.

Now let's get into the exams themselves and compare the types of sequence problems that appear. The following problem appeared in the 2016 Jiangsu Gaokao and has been translated.

Immediately we can see that this problem is much more formal than seen within SAT prep, as it defines mathematical objects using set notation. It looks like something an American college student might see in their introductory math courses. In addition to being multi-part, it requires the student to formally prove results, a concept that is largely foreign to American high schoolers. The average high school senior in the US would likely struggle to even understand what the question is asking.

To successfully solve this problem, the student must be comfortable with set theory reasoning, inequalities, geometric sequences and their sum, and proof-based thinking. Rather than solving for a numeric solution within a real-world scenario, a student must be able think abstractly about these sequences and the relationships between them.

The SAT has a very different style of problems. Let's look at a problem from a SAT practice test distributed by the College Board.

This problem presents a sequence representing a real world scenario: the population growth of a plant. It illustrates how the exam emphasizes real-world modeling and procedural execution rather than deep mathematical construction. Students are given a biological growth model and asked to apply it carefully, step-by-step. Unlike Gaokao problems, which often require students to derive and prove properties from scratch, the SAT values the ability to follow a provided structure accurately and efficiently.

Think about the average high school student. It's reasonable to assume that someone, even without much interest or specialization in math, could answer this problem correctly. It doesn't require any formal knowledge like the ones in Gaokao, and it's more about the kind of math you might use in your daily life.

A Teacher's Perspective

I decided that in addition to textbook practice and exam questions, it would be pertinent to look at tips given by Chinese teachers surrounding how to success on the Gaokao. The following tips come from a post on 知乎 (a Chinese forum similar to Quora) authored by a math teacher.

"There are not many questions that can be answered correctly by simple calculation or intuitive perception. It can almost be said that there are no questions. In order to answer the vast majority of multiple-choice questions correctly, candidates are more or less required to have certain observation, analysis and logical reasoning abilities."

"Candidates must not only provide the final conclusion, but also write or state the main steps of the solution process and provide reasonable and legal explanations."

The teacher emphasizes that while memorizing formulas and standard techniques is important, it is formal understanding, precise logical argumentation, and strategic reasoning that truly distinguish strong performers.

Students are encouraged to understand the rigorous formal meaning of every definition and property. The key is to approach problem-solving with a structured, logical mindset rather than relying solely on calculation or intuition. And in addition to being highly familiar with the formal systems within math, students are encouraged to be flexible and creative to solve problems. This includes using properties cleverly, combining algebraic and geometric thinking, and speculating based on mathematical constraints.

This real-world advice challenges a common stereotype about Chinese education being purely rote memorization. While memorizing the formulas is very helpful to achieiving speed and accuracy on the exam, it's not sufficient. High-level success on the Gaokao requires mastering a structured mode of mathematical thinking, not just memorizing facts.

Overview

In the Chinese system, preparation emphasizes formal structure, proof-based reasoning, and building solutions from first principles.
Students are trained not just to find answers, but to derive, prove, and justify mathematical relationships. Students must have mastered the topics of their high school education through years of practice and hard work.

In contrast, SAT preparation focuses on practical application, pattern recognition, and efficient problem-solving.
Students are expected to apply known formulas to real-world scenarios, often without deeply proving or generalizing them. Success depends on strategic thinking and general fluency rather than formal derivations. Students with an intuitive sense of basic math who are able to think on their feet will score highly. After all, the SAT started out as an "aptitude test"; it was meant to measure a student's aptitude for higher education aside from their high school grades.

These differences reflect two broader visions of education:

The Gaokao seeks to identify students who have properly studied a rigorous formal discipline. Although mainly future scientists, engineers, and mathematicians will make use of this specialized knowledge, the Chinese education system demands that all students pursuing higher education hold it. The mathematical thinking required will help students going into any field to think logically and carefully when problem-solving. It demonstrates that the culture as a whole values intellectual discipline and deep mastery in all educational areas.
The SAT aims to find students who can apply knowledge flexibly to everyday problems, training broadly adaptable citizens. Colleges can assume that high scoring SAT students have a strong baseline in how to understand math. In college, they will be able to take harder formal math courses without a problem, or easily incorporate math concepts into their new fields.

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