Tag Archives: education

 Henry Wan, Ph.D.

The true key to mastering mathematics lies in transforming knowledge from something external — merely memorized — into an internalized skillset that becomes second nature. To achieve this, students must go beyond rote learning and actively engage in a process of deep comprehension and creative exploration.

Step 1: Recall — Testing Your Understanding

After attending a math class, immediately review the concepts covered without referring to any materials. Try to recall the key points discussed by the teacher, including formulas, theorems, and their derivations. The ability to reconstruct this information from memory is the best test of how effectively you absorbed the material during class. If you struggle to recall certain details, it signals areas that require further review.

Step 2: Derivation — Reinforcing Knowledge Through Independent Thought

Take out a notebook and write down the important formulas and theorems from memory. However, do not stop at simply writing them down — challenge yourself to re-derive the formulas and prove the theorems using your own thought process. This independent thinking exercise helps pinpoint gaps in your understanding and solidifies your grasp of mathematical principles. The process of struggling through a derivation on your own is invaluable, as it forces you to connect different pieces of knowledge logically rather than just memorizing results.

Step 3: Comparison — Evaluating and Refining Methods

After completing your derivations, open your textbook and class notes to compare your approach with those presented in the materials or by the teacher. This comparison often leads to valuable insights: perhaps your method is more intuitive, or maybe the textbook’s approach is more elegant and universally applicable. By analyzing the strengths and weaknesses of different methods, you develop a more profound and flexible understanding of mathematical concepts.

Step 4: Generalization — Expanding Mathematical Thinking

True mastery of mathematics extends far beyond simply deriving formulas, proving theorems, or solving problems. It involves questioning assumptions, identifying patterns, and generalizing principles to uncover deeper insights. Consider the Pythagorean Theorem as an example. Memorizing, deriving, proving, and applying the Pythagorean formula for right triangles is just the starting point. To truly grasp its significance, challenge yourself with these thought-provoking questions:

• Why does the theorem specifically apply to right triangles?
• How does the relationship change if the triangle is acute? (Hint: The equation transforms into an inequality.)
• How does the relationship change if the triangle is obtuse? (Hint: The equation also transforms into an inequality.)
• How does the theorem evolve when extended from two dimensions to three, four, or eve n-dimensional space?
• What happens when we move from Euclidean space to spherical geometry? (On a sphere, the classical Pythagorean Theorem no longer holds, and an entirely new geometric relationship emerges.)

By exploring these deeper questions, you shift from merely using mathematical tools to truly understanding their foundations, limitations, and broader implications. This approach nurtures creativity, critical thinking, innovation, and problem-solving skills — hallmarks of true mathematical mastery.

A Real Example: From Competitive Math to Cutting-Edge Research

One of our students, inspired by this approach, successfully extended the Pythagorean theorem and the Law of Cosines to higher-dimensional spaces and eventually to spherical geometry. Under our guidance, the student refined this work into a rigorous mathematical paper, which we helped recommend for publication in a prestigious journal of the Mathematical Association of America. This outstanding academic achievement became a key stepping stone, ultimately securing the student’s admission to MIT.

Why This Method Works: From Passive Learning to Active Discovery

This structured learning approach — Recall, Derivation, Comparison, and Generalization — transforms passive reception into active engagement. Rather than simply memorizing formulas, you develop a deep contextual understanding and an independent problem-solving mindset. These habits not only enhance academic performance but also cultivate the ability to think critically, explore new ideas, and innovate — skills that extend far beyond the classroom.

Mathematics is not just about solving problems — it’s about discovering patterns, questioning assumptions, and pushing the boundaries of what you know. By following this learning strategy, you will not only excel in math but also develop a lifelong ability to think deeply and creatively.

 Henry Wan, Ph.D.

We have meticulously developed the Solutions Manual to provide detailed, step-by-step solutions for all homework problems. Each problem includes at least one solution, all of which were derived without the use of a calculator to encourage students for developing mathematical reasoning, problem-solving abilities, and creative thinking skills.

In cases where multiple solutions are presented, they are included to highlight key contrasts in problem-solving approaches. These alternative methods help students develop a deeper understanding of mathematical concepts by exploring different perspectives, such as:

• Algebraic vs. Geometric — Solving problems through symbolic manipulation versus visual or spatial reasoning.
• Elementary vs. Advanced — Approaching problems using fundamental techniques versus more sophisticated methods.
• Computational vs. Conceptual — Focusing on numerical calculations versus underlying theoretical principles.
• Explicit vs. Implicit — Directly solving for unknowns versus deriving solutions through indirect reasoning.
• Analytic vs. Discrete — Using continuous methods versus discrete, step-by-step reasoning.
• Forward-Solving vs. Back-Solving — Progressing logically from given information versus working backward from the desired outcome.

Understanding these different approaches helps students become more flexible and adaptable problem solvers, equipping them with the skills necessary for success in mathematical competitions and beyond.

To ensure students use the Solutions Manual as a tool for learning rather than a shortcut to answers, we have written this article, “How to Use a Solutions Manual Correctly?” This guide provides strategies to maximize the manual’s benefits while reinforcing independent problem-solving skills, ultimately fostering deeper comprehension and mathematical growth.

Many of the problems you encounter are subtle or complex, requiring careful thought — and time! — before a clear solution method emerges. The best way to learn is by attempting to solve a problem on your own, even if you don’t succeed at first. This process of grappling with challenges is crucial for deepening your understanding.

Learning mathematics requires a certain amount of “healthy frustration,” which is a natural part of developing problem-solving skills. You may need to try different approaches (some of which may lead nowhere or a dead end) until you find a viable solution. While this process can be time-consuming, it is important to persist without immediately relying on outside help. If you turn to the solutions manual too quickly, you risk missing out on the critical thinking and problem-solving skills that are key to success, particularly in competitive settings like math contests. Simply understanding the solutions provided in the manual does not mean you’ve truly mastered the material!

The solutions manual can be a helpful learning tool if used correctly, but it can also subtly hinder your progress if relied upon too heavily. How you use the manual can significantly affect your learning experience. If you use it prematurely or excessively, you may end up undermining your own efforts to master the material and perform well on contests.

The most important principle to remember is this: Do not consult the solutions manual until you have made a genuine attempt to solve the problem yourself. Ideally, you should use the manual primarily to confirm your answer. In many cases, your solution will align with the one in the manual, but occasionally, you may discover a different approach that is equally correct — or even more efficient — than the one presented. If you find an alternative method, we would love to hear about it! Please send us your solution at mathteam@ivyleaguecenter.org.

If, after a substantial effort, you still can’t find the solution, then it’s appropriate to look at solution offered in the manual. Even then, start by reading only the beginning of the solution to see if you can continue on your own. The goal is not just to arrive at the answer but to fully engage with the process, which will ultimately deepen your understanding and develop your problem-solving.

The goal is not just to reach the answer but to immerse yourself in the process, enhancing your understanding, sharpening your problem-solving skills, and fostering your creative thinking.

The goal is not just to reach the answer but to immerse yourself in the process, enhancing your understanding, sharpening your problem-solving skills, and fostering your creative thinking.

By using the solutions manual in the right way, you will maximize your learning and enhance your ability to solve problems independently — skills that are invaluable in both academic settings and math competitions.

The AMC 10 is one of the most prestigious high school Math competitions in the USA and the world. It provides a way for students interested in math to use their knowledge and skills to experience the joy of competiting against others.

The best way to prepare for the AMC 10 is to attend our AMC 10/12 training program. This is a comprehensive one-year program offering an entire course to cover all the topics that will be tested on the AMC 10/12. Students who participate in our training program will have free access to video recordings of every single class. Students who are not able to attend our training program can also purchase our course materials and recordings, including tutorial handouts, recorded videos, detailed solutions of homework problems, and mock tests, to learn all our course content. Starting each November just after the AMC 10/12 contests, we offer these sessions in sequence, as shown below.

It imust be stressed that in each session, no old material is repeated. Each session contains new, unique material, as well as a brand new comprehensive art of problem solving. We strongly believe that learning should be a long-term process of acquiring new material, information, and knowledge.

The easiest way to prepare is to practice solving previous official AMC 10 problems. Click the links below to visit our Practice Page for:

• Archived Exams: A full archive of every year’s AMC 10 A and B tests. Students can take a practice exam that includes the real questions, timing, and scoring of each exam.
• Mock Tests: Practice exams are also available for purchase. Our team developed 20 different sets of AMC 10 mock tests for students to practice with. They are intended to mimic the actual AMC 10 exam with 25 brand new questions all calibrated to be at the same style and difficulty level as the real AMC 10. In particular, they are peer-reviewed by at least two experts in math education outside our center. These simulated tests are extremely helpful for assessing students’ level of preparation for the AMC 10. Our team also devised detailed solutions to all 25 problems on our 20 sets of AMC 10 mock tests.  All these mock tests and their detailed solutions are a golden fountain of knowledge for our students, who are the ultimate beneficiaries. Surprisingly, some problems in our previous mock tests appeared in the recent official AMC 10 tests, as shown at:
  • The 2020 AMC 10/12 Contests Recycle Four Previous AIME Problems
  • Some Hard Problems on the 2017 AMC 10A/12A are Totally the Same as Previous Problems on the AMC 10/12
  • Some Problems on the 2016 AMC 10/12 are Exactly the Same as Previous AMC/ARML Problems
  • The AMC 10 and AMC 12 Have 10-15 Questions in Common

More articles about math competitions:

• AMC 10 versus AMC 12
• The AMC 10 and AMC 12 Have 10-15 Questions in Common
• Every Student Should Take Both the AMC 10A/12A and 10 B/12B!
• How the AMC contests can help you get into the Ivy League Schools
• Why are Math Competitions Important to Girls?
• Notable Achievements of Our Students
• 365-hour Project to Qualify for the AIME through the AMC 10/12 Contests
• 116 Full-length Real AMC Problems Sets are a Golden Resource to Our AMC 10/12 Prep Program
• Why Discrete Math Is Very Important
• Great Benefits of Math Competitions
• American Mathematics Contest 10 (AMC 10)
• A little competition can inspire math students to greater achievement
• Chief Instructor: Dr. Henry Wan
• 数学竞赛绝非神秘，每个学生都可参加！
• 女生更应参加数学竞赛