This article is included in the Oxbridge Essentials® “Mathematics” and “Computer Science” sections.

“Professor Xie Tao: Why prepare for the Cambridge STEP exam?”

Since 2021, when Cambridge cancelled the STEP 1 exam, retaining only STEP 2 and 3, the threshold for the exams has significantly increased. This article attempts to provide candidates with some suggestions from various aspects such as focus of preparation, time planning, training methods, and preparation resources, in hopes of helping candidates improve their efficiency in preparation and achieve desirable results in the STEP exams

## I.Key Points for STEP Preparation

The STEP exam comprehensively assesses students’ mathematical literacy and is not merely a test of the computational skills that Chinese students typically excel in. Therefore, it’s crucial for students to understand the focus of the STEP exam in order to target their training effectively and improve their STEP exam scores.

### 1. Rigor and Completeness of Mathematical Reasoning

Most students do not undergo systematic training in logical reasoning methods during their school education, which often leads them to rely on intuition when engaging in mathematical reasoning. The conditions provided in STEP problems are usually complex, and reaching the final conclusions requires several steps, where the logical relations between these steps and their rigor are significant for scoring. Complex proof problems often require discussing various cases, and incomplete discussions can lead to significant loss of marks. Teacher Xie Tao usually starts by training students in mathematical reasoning abilities, as rigor and completeness complement each other and are crucial for exam preparation.

### 2. Mathematical Proof Methods

The mathematical proof methods examined in STEP are very limited and are the most commonly used in high school mathematics. Mastering these methods themselves is not difficult; the challenge lies in choosing the appropriate method for complex problems and applying them proficiently. The application of proof methods is built on the foundation of mathematical reasoning and requires practice with a substantial number of classic problems to gain experience, thereby improving proficiency in the use of mathematical proof methods.”

### 3. Knowledge Transfer Ability in New Contexts

Many students most commonly struggle with the last sub-question of a STEP problem. As a result, many who aim for high scores in the STEP exam develop a rather utilitarian mindset: Focusing primarily on practicing the final question, learning its solution methods, and practicing enough should lead to improved scores. While it is necessary to train for the tougher final sub-questions, fundamentally it is essential to thoroughly understand the mathematical methods used in the earlier sub-questions of STEP problems and how to transfer these methods to the final question to solve the problem.

In his teaching, Teacher Xie Tao often guides students through the process of understanding and abstracting the principles from the solution processes of the initial sub-questions in classic STEP problems because only by developing this ability can one fundamentally solve the problem of transferring knowledge.

### 4. Mathematical Calculation Ability

The STEP exam does not allow the use of calculators, which means it demands a high level of mathematical calculation ability, essentially on par with the requirements for mathematical calculations in China’s national college entrance examination. Therefore, students who come from the domestic curriculum system generally meet the calculation ability requirements of the STEP exam. However, students who study international curricula such as A Level, AP, and IB often find a significant portion lacking the mathematical calculation ability to meet the requirements of the STEP exam, affecting their problem-solving speed. Hence, these students often need to undergo specialized training to improve their computational skills and speed.

### 5. Writing Standards

The writing of mathematical proofs and reasoning processes is a common weakness among Chinese students. Poor handwriting can lead directly to loss of marks, which can have serious consequences. Therefore, Teacher Xie Tao targets each student’s writing issues for specific correction.”

## II.Planning for STEP Exam Preparation

The University of Cambridge officially suggests preparing for the STEP exam for no less than 6 months. However, the actual length of preparation time varies from person to person. According to observations by Teacher Xie Tao:

Most students who achieve a grade of 1 or above in STEP usually prepare for no less than 9 months, or if less than 6 months, they spend no less than 2 hours per day on average practicing for STEP, totaling more than 400 hours of training.

For students with different mathematical backgrounds and learning experiences, Teacher Xie Tao’s recommendations are:

### 1. For Beginners in High School Mathematics Courses

For students just starting with high school mathematics courses like A Level Mathematics or AP Calculus, who usually have 12-18 months to prepare for their first STEP exam, it is recommended to practice STEP for about 5-6 hours per week, which is about half a day each week. During this period, they can also try to prepare for high-caliber mathematics competitions such as AMC and BMO.

### 2. For Students Who Have Completed A Level Mathematics or AP Calculus, But Not Advanced Mathematics

Typically, these students have 6-12 months to prepare for their first STEP exam, but with A Level final exams and AP exams coming up in June of the next year, their time for STEP preparation will be very tight. They need to fully utilize weekends and holiday time to intensify STEP training, and the earlier they start, the better.

### 3. For Students Currently Studying (or Who Have Completed) Further Mathematics and Who Have Received Conditional Offers

These students typically only have 3-6 months to prepare for their only attempt at the STEP exam, which creates a lot of pressure. There are no shortcuts in preparing for STEP; to achieve a grade of 1 or above, one must ensure more than 16 hours of practice per week and try to maximize the efficiency of preparation to avoid detours.

## III.Which Mathematics Courses Should Be Completed Before Preparing for STEP?

STEP 2 requires the content of AS Further Mathematics, while STEP 3 requires even more advanced A2 Further Mathematics content.

### 1. For Students Studying the A Level Curriculum

It is advised to start preparing for STEP after completing the entire A Level Mathematics syllabus. If you wish to start earlier, which means preparing for STEP while studying A Level Mathematics, it is recommended to begin after completing the differentiation module. It is not advisable to study Further Mathematics too early because some of the knowledge and methods, such as differential equations and hyperbolic functions, can be easily forgotten if not used for a long time. Therefore, it is generally recommended to start preparing for STEP while studying Further Mathematics to promote mutual reinforcement, maximizing learning and preparation efficiency.

### 2. For Students Studying the AP Curriculum

You can start preparing for STEP when you are about halfway through AP Calculus, but make sure to allocate time to supplement knowledge from the pure mathematics part of A Level Further Mathematics. The knowledge from AP Physics C Mechanics basically covers the mechanics topics in STEP, so it’s not necessary to study the mechanics part of Further Mathematics specially. However, AP Statistics differs significantly from STEP Statistics, so if you wish to try STEP Statistics questions, it’s still necessary to supplement with advanced statistical knowledge from Further Mathematics.

### 3. For Students Studying the IB Curriculum

Based on past cases, students from the IB system usually do not have an advantage when preparing for STEP, mainly because they face a lot of pressure from their regular coursework, which compresses their preparation time for STEP. It is recommended that IB students complete both SL and HL Mathematics early and allow for an extended period for STEP preparation.”

## IV.Methods for Preparing for the STEP Exam

There are two main phases or methods of preparation: systematic study and practice with past papers. However, it is highly discouraged to rely on practicing past papers alone for STEP exam preparation.

### 1.Why isn’t it advisable to replace systematic learning with practice from past papers?

Many students, due to time constraints, hope to apply the successful experiences of preparing for A Level Math, AP Calculus, or IB Higher Level Math (hereinafter collectively referred to as international high school math)—by practicing a few sets of past papers to see what topics have been covered in previous years, then practicing a few similar problems to achieve high scores. The following comparison explains why this approach is not suitable for preparing for the STEP exam:

In summary, before entering the practice phase, students need to ensure they have enough time to systematically learn and master the knowledge and methods required for the STEP exam.

## ii. How to Conduct Systematic Training?

Mr. Xie Tao has distilled years of STEP teaching experience into a systematic training approach that can be summarized in five points: emphasizing mathematical knowledge, thought processes, calculation speed, problem-solving techniques, and writing habits equally.

1. Completing the Mathematical Knowledge System Whether it’s A Level Math, Advanced Math, AP Calculus, or IB HL Math, the content doesn’t completely cover the STEP syllabus. Therefore, supplementing the study of content beyond the scope of these curricula is a vital part of the STEP preparation process.

2. Training Mathematical Thinking Methods In teaching practice, many students are naturally talented in mathematics but may jump to conclusions or lack rigorous logical reasoning. Some also lack systematic methods of mathematical thinking, all of which can impact STEP problem solving and scoring. Generally, after systematic and targeted guidance, issues related to critical mathematical thinking methods tend to see significant improvement.

3. Independence from Calculators and Improving Calculation Speed International mathematics curricula allow the use of calculators, especially for students studying in American high schools or AP courses, who rely heavily on the computational and graphing capabilities of graphic calculators. This dependence results in some students struggling with mental arithmetic and slow problem-solving speeds when calculators are not allowed, as in STEP exams. The computational workload of the STEP far exceeds that of A Level and Further Mathematics, and calculators are not permitted. Therefore, students preparing for STEP should minimize or avoid using calculators and practice persistently to naturally enhance their calculation speed.

4. Accumulating Problem-Solving Techniques Through Practice Many STEP problems come with novel methods that can be easily learned by carefully observing and pondering the intentions behind the questions. Students can often improve their mathematical thinking by reviewing and reflecting on problems under a teacher’s guidance. Long-term practice accumulates a variety of commonly used problem-solving techniques. Integrating these techniques boosts mathematical literacy and increases the chances of becoming a preferred candidate for universities like Oxford and Cambridge.

5. Developing Good Writing Habits STEP emphasizes the rigor of logical reasoning, and a standard writing process with concise textual reasoning showcases a solid mathematical foundation. Irregular writing and expression often become obstacles to achieving high scores in STEP exams. Writing habits cannot be developed overnight; strict requirements and diligent practice in organizing and expressing mathematical language must start from the beginning of STEP preparation.

Each student’s personality and talent are unique. When developing a systematic training plan, it’s essential to combine the actual situation of the student with long-term planning to achieve the best preparation outcomes and ideal STEP scores with the least time and effort.

## iii. Ensure to Schedule Mock Exams

Despite the conflict of STEP exam dates with other significant exams, it is still recommended to schedule 2-3 STEP mock exams. Typically, a past paper should be attempted about 3-4 weeks before the exam, the second set around 2 weeks before, and then another set around 3 days prior.

The mock exams should use recent papers from the last three years to get a feel for the recent trends in STEP question setting and to adapt to the difficulty level of the STEP exam.

During the mock exams, it’s crucial to strictly adhere to the timing requirements and to score against the grading criteria.

After each mock exam, it is essential to promptly summarize, identify knowledge gaps, and then practice targeted exercises for consolidation.

## V. Recommended Preparation Resources

### 1. Collection of Past Papers

Click on the images below to browse through the collection of past papers for STEP 2 and STEP 3.

Youth Education Consulting | Cambridge STEP 2 Past Papers Collection (1987-2023) Youth Education Consulting | Cambridge STEP 3 Past Papers Collection (1987-2023)

##### Click here to download the PDF collection for free>>

The above two sets of past paper collections (PDF) have organized all the questions from STEP 2 and STEP 3 since 1987, with clear text and beautiful design, convenient for students to consult and use for mock exam practice.

### 2. Xie Tao’s STEP Preparation Series Courses

STEP 2023 [6th Edition] course is adapted to the latest 2023 syllabus. This edition has added some new examples and exercises, slightly increasing the difficulty level of the exercises. Click on the image link to view course details.

### 3. Cambridge STEP Preparation Support Programme

The official Cambridge STEP support site offers free STEP Foundation, STEP 2, and STEP 3 assignments and mixed questions. Each assignment comes with an answer. This set of materials is somewhat helpful for STEP beginners. The downside is that there are few questions, with only about four per assignment, and the questions are relatively easy, so students with a higher level can complete them in about 1-2 hours. To achieve a grade 1 or even an S, this material alone is far from sufficient.

### 4. Officially Recommended Problem-Solving and Textbook Readings

#### How to Think Like a Mathematician

Published by Cambridge University, this book explains basic mathematical thoughts, logical reasoning, and proof methods. Most chapters come with a few exercises. Despite being recommended for undergraduate math majors, it is also suitable for high school students preparing for mathematics majors at Oxford or Cambridge.

#### Solving Mathematical Problems

Published by Oxford University, this book is less than 100 pages and introduces some basic strategies for solving mathematical problems, including easily understandable example problems that cover number theory, algebra, Euclidean geometry, and analytic geometry.

#### Advanced Problems in Mathematics

Officially recommended by Cambridge for STEP exam preparation. If these books are not challenging enough, you might try the one below.

#### A Concise Introduction to Pure Mathematics

Some of the examples and methods in this book are basically prototypes of STEP exam questions.

If you still find these materials insufficient, refer to “Application Guide | Mathematics Majors at Oxford and Cambridge” for more officially recommended readings.

### 5. Comparable Level Mathematical Olympiad Problems

British Mathematical Olympiad (BMO) is recommended. BMO Round 1 consists entirely of short-answer questions (in 2020, due to the pandemic, the format was changed to include multiple-choice and short-answer questions). Some of the questions related to algebra, sequences, and geometry are similar to the style of Cambridge STEP questions and can be used for practice. It should be noted, however, that high school math competition problems generally do not cover calculus, which is a must for the Cambridge STEP exam and constitutes a significant portion.

For students who start preparing for the STEP exam in the first semester of 10th grade (G grade) or 11th grade (AS level), American mathematics competitions like AMC 10 and 12, and British math challenges like UKMT’s IMC and SMC are also highly recommended.