One day in March 2025, I was watching one of my friends study biology. In the process, they taught me some cool things. But then I had an idea: what if I taught them calculus in return?
We proceeded to have a very fruitful discussion about infinite series, and we both had a lot of fun along the way. My friend felt the joy of discovering and learning new concepts, and I felt the joy of spreading my knowledge. In the weeks following this, we had a few more sessions together, with us exploring more topics related to series and integration.
What is an infinite series?
Infinite series concern the summation of an infinite number of terms. I started off my discussion with the following series:
How does it make sense to sum an infinite number of terms? If you don’t already know about series, pause and think about this for a moment, then click the button below when you’re ready.
Since we can’t add up infinitely many numbers in the most literal sense, we can see what happens as we add more and more of them:
What number does the sum approach as we add more terms?
As we add more terms, the sum approaches 1, meaning that it makes sense to say that the sum of all infinitely many terms is equal to 1.
An image demonstrating the summation of this series. The entire square has an area of 1.
After my discussion of this series, I had my friend think about two more series:
I’ll let you ponder the same questions I asked my friend: Does this method of adding more and more terms and seeing what the sum approaches work for these two series? If not, what goes wrong?
Fast forward a few weeks later, and I decided to attend this friend’s calculus class out of boredom. The professor went over the Fundamental Theorem of Calculus, which (as its name suggests) should be one of the most exciting moments in the entire course, the moment when two of the main concepts in calculus, derivatives and integrals, come together.
The Fundamental Theorem of Calculus
(This is optional; feel free to skip this section if you don’t understand the concepts.)
Most of the time in a calculus class is spent studying derivatives and integrals. A definite integral gives the area under a function’s graph. A derivative tells us how much a function’s value changes as you change its input by a tiny amount. Are these two ideas related, and if so, how?
Imagine a function \(F\) that tells you the area under a function \(f\) from some starting point (say \(x = 0\)) to a given \(x\)-value. Now imagine increasing the input of \(F\) by some tiny amount. How will the area change?
How much does the area function \(F(x)\) change by when \(x\) increases by some tiny amount \(\Delta x\)? An approximation is fine.
The area added in the diagram above is approximately \(f(x)\Delta x\). In addition, this is an approximation that gets better and better as \(\Delta x\) shrinks.
What this means is that the rate of change at which \(F(x)\) changes (the change in \(F(x)\) divided by the change in \(x\)) is equal to \(f(x)\Delta x/\Delta x\), which is equal to \(f(x)\). In other words, the derivative of \(F(x)\) is \(f(x)\). This is important because it tells us that the derivative of an area function gives us the original function, which means that derivatives and integrals are inverses! This fact is known as the Fundamental Theorem of Calculus.
Yet what actually happened was the opposite. My friend was completely disengaged. There was no sense of exploration or excitement as we watched the professor give us the formula and work out problems. After one of the other lectures I attended, my friend told me that it felt like a “nothing burger.”
The contrast between the lecture and our session together is striking to me. How could this be possible? What has gone wrong, and does learning mathematics have to be this way?
The area of a triangle
Believe it or not, some of the problems with mathematics education can be revealed simply by thinking about a triangle in a box. What is this triangle’s area relative to the rectangle that surrounds it?
How much of the box does this triangle take up?
If you don’t already know the answer, play around with it using the slider and see what observations you notice before continuing.
You might have noticed that if you turn the triangle into a right triangle (by moving the slider to either end), the triangle obviously takes up half the rectangle. Is this true in general?
The answer becomes a lot clearer when we add a vertical line across the altitude of the triangle:
Notice how the vertical line splits the rectangle into two regions (one to the left of the line and one to the right). The triangle takes up exactly half of each region, meaning that the triangle always takes up half of the area of the rectangle.
As Paul Lockhart says in his Mathematician’s Lament, this is a simple yet beautiful example of how a mathematician would solve a problem: by playing around and trying things until they come to a solution. Yet in schools, students don’t get a chance to explore this problem and discover the answer themselves. Instead, they are just given this formula:
\[ A = \frac{1}{2}bh \]
Instead of students seeing this piece of mathematics as an interesting puzzle and an opportunity to use their problem-solving skills, they are presented with yet another thing to memorize.
This is strikingly similar to the situation I mentioned at the beginning: instead of my friend getting a chance to discover the Fundamental Theorem of Calculus, one of the most important theorems in all of calculus, they were simply handed a formula to memorize.
The sum of odd integers
Take a moment to calculate these sums of odd numbers.
Do you notice an interesting pattern? Can you find an expression for the sum of the first \(n\) positive odd integers?
The sums turn out to be the perfect squares (1, 4, 9, 16, and so on)! In general, the sum of the first \(n\) positive odd integers is \(n^2\).
Notice how I allowed you to explore the problem for yourself first, hiding the solution behind a button (also note how I’ve already done this twice before on this page). I could have just directly told you as a bland statement of fact that the sum of the first \(n\) odd numbers is \(n^2\), but don’t you think it’s much more engaging to discover this fact yourself?
This problem is also a simple one that even a young student could solve, showing that this type of mathematical thinking and exploration isn’t restricted to the “smartest” students or those with lots of math experience.
As a bonus, try to think about why this fact is true. Can you think of a reason why adding odd numbers gives the perfect squares?
For an extra challenge, try to find the sum of the first \(n\) even integers. Can these results help you find a formula for the first \(n\) positive integers?
Some of my personal experiences
Throughout my years of schooling, I’ve often felt like math class is just a list of things to memorize, often with the true importance or intuition of the concepts being unclear. Here are just a few examples (reading them is optional):
Radian angle measure
The input area below will calculate the sine of an angle you enter. Try entering a few small angles here (like 0.1 degrees or 0.01 degrees). What do you notice about the sine of these angles?
Enter an angle: degrees
When you calculate the sine of a small number, the result is approximately equal to the angle you entered multiplied by 0.01745. In other words, for small values of \(x\):
\[ \sin(x) \approx 0.01745x \quad (\text{when }x\text{ is in degrees}) \]
But where does this strange number come from, and does it have an important meaning? It turns out that this number is just an artifact caused by the way we measure angles in degrees.
Wouldn’t it be nice if for small values of \(x\), \(\sin(x)\) was approximately equal to \(x\)? Let’s take a look at what sine represents to see if we can make this happen:
The angle \(\theta\) depicted on a unit circle.
\(\theta\) in this diagram represents a small angle. What do you notice about the arc length that subtends the angle \(\theta\) (the red arc length in the diagram) and the value of \(\sin(\theta)\) (represented by the length of the blue line segment)?
The arc length that subtends \(\theta\) is very close to the value of \(\sin(\theta)\), and this approximation only gets better as \(\theta\) gets smaller and smaller. So maybe it would be useful to design a system of measuring angles that is based on this arc length. We could say that 1 radian is the angle that an arc length of 1 subtends in a unit circle.
This would mean that an arc length of \(x\) subtends an angle of \(x\), and since this arc length is about \(\sin(x)\) for small \(x\), we would have that \(\sin(x) \approx x\) for small values of \(x\).
This fact is very useful in calculus and makes a lot of calculus facts simpler, such as the fact that the derivative of sine is cosine (this is only true in radians).
In radians, as a result of the small angle approximation \(\sin(x) \approx x\), the slope of the sine graph at \(x = 0\) is 1. In degrees, the slope at that point is the same number we found earlier, about 0.01745. This number is actually the number of radians in a degree, \(\pi / 180\).
The graph of \(\sin(x)\), where \(x\) is in radians. The slope at \(x = 0\) is 1.
The graph of \(\sin(x)\), where \(x\) is in degrees. The slope at \(x = 0\) is about 0.01745.
When I learned about radians for the first time, I was never given a reason why it would be a good idea to switch our system of measuring angles, and to a lot of students, it probably felt like we were learning a new system for no reason or just to make mathematics unnecessarily complicated. However, radians actually make things much simpler in a lot of cases!
Logarithm properties
Imagine a bank that doubles your money each year. For example, if you put $1 in, you will have $2 in 1 year, $4 in 2 years, and so on.
Time elapsed (years)
Money (dollars)
0
1
1
2
2
4
3
8
4
16
If we start with $1, we can express the amount of money we have after \(t\) years using the exponential function \(M = 2^t\). But how can we write the amount of time elapsed as a function of how much money we have?
We can use a new type of function, a logarithmic function, to model this. In this case, we can say that \(t = \log_2(M)\). This is the inverse of the exponential function \(M = 2^t\), since instead of going from time elapsed to account balance, we are going from account balance to time elapsed.
In other words, the \(\log_2\) function takes in an amount of money and tells us how long it takes to get from $1 to that amount (assuming our money doubles each year). Equivalently, the \(\log_2\) of a number \(x\) tells us how many years it takes to multiply our money by \(x\) (for example, it takes us 3 years to multiply our money by 8, so \(\log_2(8) = 3\)).
The expression \(2^t\) takes in an amount of time and tells us how much our money is multiplied by. The expression \(\log_2(M)\) takes in a number and tells us how much time it takes for our money to be multiplied by that number.
Let’s say it takes \(a\) years to multiply your money by 3 in this bank and \(b\) years to multiply your money by 7. How long will it take for you to multiply your money by 21? What does the answer to this question tell us about the values of \(\log_2(3)\), \(\log_2(7)\), and \(\log_2(21)\)?
To multiply your money by 21, you first multiply your money by 3 (which takes \(a\) years) then multiply your money by 7 (which takes \(b\) years). In total, it takes \(a + b\) years to multiply your money by 21.
This means that \(\log_2(21) = \log_2(3) + \log_2(7)\). Note that this is true not just for this particular bank that doubles our money every year, but for any bank with a consistent interest rate, meaning that this identity holds for any logarithm base!
This also works for numbers besides 3 and 7; by following similar logic, we can derive the identity
\[ \log_c(ab) = \log_c(a) + \log_c(b) \]
where \(c\) is any valid logarithm base.
In my personal experience, I just memorized this formula in Algebra II without putting much thought into why it worked. In the following years, would use this formula many times when doing math, treating it as no more than a rule I needed to apply when I was working with logarithms.
But as I was writing this article, I came up with this method to justify the logarithm properties, and this made it so much more clear to me why they works. There’s almost nothing to memorize now - just remember that logarithms tell you how long it takes to multiply your money by a given amount, and the logarithm properties become almost obvious!
As a bonus, if it takes \(a\) years to multiply your money by 3, how long does it take to multiply your money by \(3^{10}\)? What does this tell us about the values of \(\log_c(3)\) and \(\log_c(3^{10})\), where \(c\) is any valid logarithm base?
Does the same logic work if the numbers 3 and 10 are replaced with any other numbers? Can you use this to find an identity for \(\log_c(a^b)\)?
The number \(e\) and the natural logarithm
Imagine a bank that offers a very special deal: at any point in time, the amount of interest it gives per year is exactly equal to your account balance. For example, if you have $100 in the bank, it will give you exactly $100 per year in interest at that point in time.
You put $1 in the bank and after a year, the bank gives you $1 of interest, bringing your account balance to $2.
However, you notice something interesting: the bank hasn’t quite fulfilled its promise.
The bank has maintained an interest rate of $1/year throughout the whole year, but your account balance has been greater than $1 for almost the whole year! As soon as you earned even a tiny bit of interest, your interest rate should also have been updated to reflect this.
So the bank decides to simulate what should have happened, but instead of keeping the interest rate at $1/year for the whole year, it updates your interest rate halfway through the year from $1/year to $1.50/year. By doing this, you now end up with $2.25 at the end of the year (since you started with $1, earned $0.50 in the first 6 months, then earned $0.75 in the last 6 months with the increased interest rate).
However, you’re still not satisfied, because for the first half year your interest rate was lagging behind your bank account balance.
So to stop you from complaining again, the bank then tries to see what happens as it updates your interest rate to match your account balance more and more frequently. What do you think will happen? How much money do you think you will end up with after 1 year?
Use this slider to find out what happens as the bank updates the interest rate more and more frequently:
The red line represents your account balance over time and the blue line represents the interest rate you’re being paid. What happens as the bank updates your interest rate more and more frequently?
As the bank updates your interest rate more and more frequently, the amount of money you end up with after 1 year approaches about 2.71828. This number is known as Euler’s number and is denoted by \(e\).
In this bank scenario, the amount of money in your bank account as a function of time \(t\) in years is \(e^t\).
Now obviously in the real world you won’t find a bank that gives you interest equal to your account balance (that would be a 100% interest rate)! Nevertheless, there are many real-world scenarios that involve a value that is changing at a rate proportional to itself, such as population growth and radioactive decay. In these scenarios, the number \(e\) will naturally arise when trying to write this value as a function of time.
For example, if a value is continuously growing at a rate equal to 20% of itself, the value as a function of time can be represented by a function of the form \(f(t) = Ce^{0.2t}\). (If you know calculus, try finding the derivative of this function with respect to \(t\). What do you get? Is its derivative really equal to 20% of the original function?)
When I first learned about the number \(e\) in Algebra II, I was confused on why it was given so much importance. The only explanation I got is that it had something to do with continuously compounded interest, and I was oblivious to why it shows up so often in mathematics - it appears any time you have a value that is continuously growing at a rate proportional to itself!
The natural logarithm \(\ln(x)\) is the inverse of the exponential function \(e^x\), and it tells us how long it takes to multiply our money by \(x\) in this magical bank where our interest rate is always equal to our account balance.
In general, whenever you have a value that’s growing according to a function of the form \(f(t) = Ce^{rt}\) (which appears when you have a value that’s changing at a rate of \(r\) times itself), the natural logarithm will help you determine how long it will take for that value to reach a certain number.
When I learned about the natural logarithm, I was even more confused about why we would ever use the natural logarithm as opposed to logarithms of other bases. It wasn’t until I learned calculus that I truly understood the natural logarithm and the properties that made it so useful!
What can we do about this?
Despite these issues, there are still steps we can take to improve students’ learning experiences. When we’re helping students with math individually:
We can make sure to go beyond computations and formulas and explain the underlying concepts in more depth.
We can teach with an intent of strengthening students’ understanding and not just to help them get better grades.
We can allow students to have a greater role in their learning, allowing them to discover important facts rather than simply telling them the results.
We can make mathematics feel more like an engaging puzzle rather than a list of facts and procedures to memorize - a way to discover surprising relationships between ideas and find solutions to some of humanity’s most interesting abstract problems.
(For more tips on helping students with math, visit my tutoring guide.)
I don’t blame teachers and professors for the state of mathematics education, as there isn’t much you can do to make learning more interactive when you’re teaching dozens or even hundreds of students at the same time. These techniques are best used for one-on-one or small group sessions.
But if you’re a student or tutor, you have the power to make learning math better for your fellow peers! You have the power to share the beauty of mathematics with others! Even a short discussion about mathematics with one of your friends can make the subject less stressful and more exciting.
Many students hate mathematics not because the subject is tedious and meaningless, but because they’ve never seen math presented in an engaging way. You have the power to change this!
“Personally, I am immensely indebted to many brilliant, wonderful educators, who opened my mind, broadened my horizons and offered guidance that I sorely needed. But one quality that they all shared was a deep respect for the learner, and an understanding that education takes two – it is a dialogue, not a monologue. All of them were discontented with the system that they had to work within and often subverted it simply by being excellent teachers.” - Franklin Liu (one of my high school friends). Read his essay about education here: Education is Broken. Only We Can Fix It.
Before you say “I’m not good enough at math to help others with it,” think for a moment about what mathematics is to you. Do you see mathematics as a set of rules and procedures to be memorized, or do you see it as the creative art of discovering abstract patterns and solving problems using them?
Mathematics is beautiful and culturally important, and we should treat it not just as a science but also as an art. Every student deserves the chance to discover the wonders of mathematics. Can you imagine an art class where you never get the chance to create your own works, or a music class where you never listen to any pieces?
