calculus gaming’s tutoring guide

By Eldrick Chen, creator of calculusgaming.com

Originally published on 2025-08-15

Introduction

Here are some of the tips I have for being an effective math tutor. Note that most of these strategies come from what I’ve found to be effective in my own experience, and different people might have different preferred ways of tutoring. Tutoring is just as much an art as it is a science, and there is no one universally best way to teach any concept.

Who this guide is for

These tips are useful if you’re individually tutoring a student or if you’re helping one of your friends/classmates with homework. I won’t be going over tips for teaching multiple people at the same time or teaching in a classroom setting. I’ll also assume that the people you’re tutoring are fairly mature, so I won’t go over disciplinary tips.

After all, part of the reason I made this guide is for myself, and I primarily tutor college students individually. If you’re a parent or classroom teacher, there are lots of other resources out there on how to teach children (nevertheless, you might still find some of my tips useful).

This guide will focus specifically on tutoring mathematics, but many of the tips are also applicable to other subjects.

One final note: I’m not a professional tutor or teacher. I’m just a college student who loves teaching math to my friends and other students.

Explaining concepts

A good amount of time tutoring math will be spent explaining topics that the student is trying to learn. Here are some tips I have for this aspect of tutoring.

Understand your students’ current level of understanding

Before you start tutoring, ask a few questions to understand where your students are at. Then, adjust your explanations appropriately depending on how much they know - avoid using too much jargon when teaching those who don’t have a strong understanding of the material.

Example: Imagine you’re taking your first calculus class ever and you don’t have a strong understanding of limits yet. Which of these explanations would be more useful to you?

Explanation 1: The limit of a function \(f(x)\) as \(x\) approaches \(c\) is defined as the real number \(L\) such that for any epsilon greater than 0, there exists a delta greater than 0 such that for all \(x\) not equal to \(c\), the absolute value of \(x - c\) being less than delta implies that the absolute value of \(f(x) - L\) is less than epsilon.

Explanation 2: The limit of a function \(f(x)\) as \(x\) approaches \(c\) is the value that \(f(x)\) gets closer and closer to as \(x\) gets closer and closer to \(c\).

You would probably find Explanation 2 much easier to understand. In contrast, you would likely feel overwhelmed by Explanation 1 and would likely end up not understanding any part of it.

This doesn’t mean that Explanation 2 is inherently better than Explanation 1; the first explanation is much more precise and rigorous and would be more insightful later on in a calculus course. However, Explanation 1 is only effective when the student already has a strong understanding of limits and is very comfortable with mathematical notation, so using Explanation 2 is much better for those who are learning the concept for the first time.

Encourage the student to think and discover

If time allows, don’t simply tell the student about mathematical results, but instead let them explore and discover the concepts themselves. This will lead to a much more engaging experience overall, where the student is discovering new things instead of just memorizing formulas.

Example: Imagine you’re learning algebra. I could directly tell you that \((a + b)^2 = a^2 + 2ab + b^2\), but then you might be tempted to just memorize this formula without an understanding of where it comes from or why it’s true.

I could improve this by giving you a quick explanation of where the formula comes from, but this still doesn’t allow you to do much thinking on your own.

Instead, I could ask you to draw a visual representation of \((a+b)^2\) and use that diagram to find an alternative representation. After a bit of tinkering (and a little bit of guidance from me if you get stuck), you might come up with this diagram:

This square has a side length of \(a+b\), so the area of the square is \((a+b)^2\). The combined areas of the 4 regions is \(a^2 + 2ab + b^2\). Therefore, \((a+b)^2 = a^2 + 2ab + b^2\).

The key is that I allowed you to do most of the thinking (not me!), which led you to discover the result.

By doing this, you end up learning a lot more than if I had simply told you the formula, and I’ve also trained you to think like a mathematician!

Explain concepts in multiple ways if possible

Often in math, there are multiple ways to explain the same concept. Sometimes a student might have a hard time understanding one particular explanation of a concept, so try to switch to a different explanation if this happens.

Example: Let’s say you wanted to teach a student about the value of the infinite series \(1/2 + 1/4 + 1/8 + \cdots\). There are multiple ways you could approach this.

You could use an algebraic argument by showing that the following pattern holds:

\[ \begin{alignat}{1} & \frac{1}{2} \,\,&&= \frac{1}{2} \,\,&&= 1 - \frac{1}{2}\\ & \frac{1}{2} + \frac{1}{4} \,\,&&= \frac{3}{4} \,\,&&= 1 - \frac{1}{4}\\ & \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \,\,&&= \frac{7}{8} \,\,&&= 1 - \frac{1}{8}\\ & \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \,\,&&= \frac{15}{16} \,\,&&= 1 - \frac{1}{16}\\ & && \vdots \end{alignat} \]

Because the partial sums are approaching 1, the sum of the series is equal to 1.

Alternatively, you could use a geometric approach:

The area of the square is 1. Because the rectangles with area \(1/2\), \(1/4\), etc. fill up the entire square when extended to infinity, the sum of the series \(1/2 + 1/4 + 1/8 + \cdots\) is equal to 1.

Bonus points if you used the previous strategy (“Encourage the student to think and discover”) and gave the student a chance to discover one of these explanations for themselves!

Check for understanding

Don’t move on to a new concept until the student has understood your explanations. Ask specific questions to ensure they understand what you’ve said, or alternatively, have them re-explain the concept to you. Try not to just ask if the student understands what you said as they might not give you an accurate answer.

Examples of questions to avoid:

Examples of stronger questions:

(This is something I personally have to work on; I’m guilty of often asking simply if students understand!)

Ask open-ended questions

Whether you’re testing a student’s knowledge or helping them explore a new concept, open-ended questions can be a powerful tool to keep them engaged and get them to think. Avoid asking only yes/no questions (or questions with a very limited range of answers); be sure to ask a good number of thought-provoking questions that could lead to interesting discussion.

Examples of simple questions (only a few possible answers):

Asking questions like these is fine, but you should also mix in open-ended questions when tutoring.

Examples of open-ended questions:

Sometimes simple questions can be improved simply by asking why a student answered the way they did. For example, “Is 17 a prime number?” becomes “Is 17 a prime number? If so, why?”

These open-ended questions can be tough, so remind your students that it’s okay to get the wrong answer - this is a natural part of learning.

Be flexible and understand the material very well

It’s okay to have a lesson plan for each tutoring session, but keep in mind that things might not go as planned. It’s important to be able to deal with unexpected questions or difficulties, and in general, you need the skills to tutor on the spot. Be sure to have a very strong understanding of the material so you can efficiently resolve any issues with the students’ understanding.

Example: You’re teaching a student about the derivatives of common functions, and you say that \(e^x\) is its own derivative. The student then asks you why this is true.

Without a strong understanding of the concept, it can be hard to answer this question. One common answer is that \(e\) is defined as the number such that \(e^x\) is its own derivative.

Your student might bring up another definition of \(e\) they learned: \(e\) is the limit of \((1+1/n)^n\) as \(n\) approaches infinity. They then ask you how this definition relates to \(e^x\) being its own derivative.

This question is even harder to answer and requires an even stronger understanding of the number \(e\).

Try to personalize your explanations

If possible, try to create explanations that your student might be able to relate to. You can do this by asking what your students are interested in, then designing explanations that relate to their interests. This one might be hard to achieve as students have a wide range of interests and you might not know enough about these interests to connect the material to them.

Solving problems

Here are some tips to use when students are solving specific problems.

Don’t give answers directly

You should avoid giving answers to problems directly, as the student might not learn much this way. Instead, try to give hints to guide the student towards the right answer. Design your hints in a way such that the student still has to do a decent amount of thinking to get to the answer.

Another way of saying this is that you should emphasize the process of getting an answer over the answer itself so that the student will know how to solve similar problems in the future.

Example: Pretend you’re learning about integrals for the first time. If I told you that the definite integral of \(x^2\) from \(x = 0\) to \(x = 2\) was \(8/3\) but I didn’t tell you why (or how you would figure that out yourself), have you truly learned anything?

Instead, I might encourage you to try finding a function whose derivative is \(x^2\), then think about how this function could help you find the definite integral in question.

Reinforce the correct process

When a student gets an answer correct or incorrect, try not to simply say “That’s right” or “That’s wrong”; instead, ask them how they arrived at their answer. This can help the student understand why they got it correct or incorrect, and can help you notice when the student is guessing instead of actually understanding.

Here’s a fictional conversation that demonstrates this technique.

Tutor: What is the derivative of \(x^2 + \sin(x)\)?

Student: The derivative is \(2x + \cos(x)\).

Tutor: How did you get this answer?

Student: I used the power rule to differentiate \(x^2\) and I know the derivative of sine is cosine. In addition, the derivative of the sum of two functions is the sum of the two functions’ derivatives.

Tutor: That’s correct!

Be active, but don’t be too active

As a tutor, you want to be active in the student’s learning process, but you also want to give the student enough time to think for themselves. If you as a tutor are explaining concepts without letting the student take time to think through them, they might end up not learning as much. You also don’t want to overwhelm the student by providing too much information at once. You want to give students adequate time to work through problems on their own.

Creating a safe learning environment

Positive feedback

Believe in your students! You want to encourage the student to keep learning and not give up. One way of doing this is by using positive feedback: by acknowledging what the student has done right. However, you should be careful: avoid praising inherent attributes of a person and instead praise specific things that the student has done well. Ideally, you want the student to attribute their success to their own hard work.

Example: Imagine a student has answered a question correctly, and you praise them for being smart. However, they proceed to answer the next question incorrectly. Now the student might think that they got it wrong because they weren’t smart.

Instead, you could praise the student for not giving up and thoughtfully working through each step. This will lead the student to associate success with their own hard work, and might encourage them to keep trying hard.

Allow for mistakes

Students are often afraid of making mistakes, but mistakes are an essential part to learning. Allow your students to make mistakes and tell them that it’s perfectly fine. Analyzing errors can also help your students and teach them to not make the same mistake the next time. Doing this will also encourage the student to try new things and perhaps discover something they hadn’t known before!

Be patient

Sometimes a student will take longer than expected to understand a concept. You should be fully prepared for this as different students have different levels of understanding at a particular time. In this case, try not to appear frustrated or impatient, as that can cause students to feel like they’ve done something wrong.

Sometimes students might struggle with concepts that are simple to you; remember that not everyone is at the same level of understanding as you! It can be frustrating to see (for example) calculus students struggle with basic algebra, but as a tutor you should be prepared for this since this is (unfortunately) not a rare occurrence at all.

Be human and show passion

Avoid giving the impression that you’re just reading off a script you memorized. Try to have genuine conversations with your students and don’t be too formal. Don’t be afraid to show your personality! In addition, try your best to show that you’re genuinely interested in the material and want your student to understand it very well. Tutoring doesn’t have to be boring, so have fun!

Encourage students to ask questions

A large part of the learning process is asking questions. As a student myself, sometimes when I’m in class, I’m afraid to ask questions because I don’t want to waste everyone’s time. In a one-on-one tutoring session, this isn’t as much of a problem, so let your students ask questions and even encourage them to do so!