Math, Calculus, and Beyond

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Short Answer

“It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.”

  • Henri Poincaré, Science and Method (2011)

Notice: This entire chapter is, at best, a surface level view of each topic covered here. That is intentional. The next few chapters are separated to avoid spilling over material and causing undue confusion.

If there’s one thing you at least need to take away from this chapter, it’s people should memorize their times tables.

The average adult (and people becoming adults) should have math skills. I’m not expecting the average person to solve the P vs NP problem or another Millennium Prize Problem on their own, but I am expecting the average person to know how to handle math in their future.

  • Let me stress once again: every adult should care about math and have some knowledge in it.
  • Put another way: If you need to commute, you just need to know how to drive a car.
    • Alternatively: At a minimum, learn the high-level concepts.

I have bias writing this chapter because my background in education is as a Science/STEM Teacher (and substitute teacher). This chapter, however, covers math and some of its applications, based on what I use outside education systems and as part of a graduate engineering degree (M.S.I.E.).

  • Additionally, most Science in general requires the student proficient in math.
  • Even if you forget most the math here in 5-10 years after initially learning, you’ll still (hopefully) retain logical thinking and reasoning habits acquired during the learning process.

I believe learning mathematics isn’t optional anymore and absolutely necessary. In the Information Era, and projected future of humanity, understanding math is like a survival skill. A lot of math you may see was also made precisely to solve practical problems too, like Calculus for orbital mechanics, so it’s pretty cool to know about and see in action.

You don’t need to be an expert in mathematics, know how to do complex math well, or even expect you’ll see an application for all math theories and methods. You should, however, acquire familiarity with the abstract and concepts behind the math to prove properties/things, recognize what is happening, and when math is used for deception.

Lastly, you’ll want advanced knowledge precisely to know when an advanced method is not the best course of action and to know how to manipulate advanced methods to better suit your needs.

Long Answer

Math is about theory and proofs.

Science is application of math to solve real problems.

Applying math is what makes money.

Money (and currency) affords goods and services to keep you alive.

Money or no money, doing cool things with math, even if you cannot directly apply it, is still pretty cool. The less cool part, though, is the field is so incredibly diverse that no one single person will likely ever learn all of it.

The average person may not get much mileage out of math, but the smarter person may. Not every person is expected to become an engineer (or even complete an engineering degree) or suddenly be good at math just because they took a course in it. There’s also adjacent roles using math like physicists and analysts, and countless roles using math in some way, shape, or form.

If I had to name only two issues affecting math, it’d be the following:

  1. Math perceived negatively is normalized, so being bad at math is “ok” and competency starts sliding down.
  2. How math is taught across the entire learning sequence and all related curricula is flawed.

I’m not sure which of these is easier to fix, as both issues could be forever ongoing or take beyond my lifetime to fix up. I’m not here to write a full solution to that because I’m focusing on the importance of math, which means more targeting issue 1 over issue 2.

The main thing stopping people is math is “difficult,” but not in ways you’d expect, and it still remains difficult even under perfect conditions. Sometimes it’s so difficult even the professor or teacher outright admits the subject is just that intuitively difficult or unable to verbalize well before attempting to teach it anyways. There are certain fields where you can learn enough to explain what you don’t understand. Due to this difficulty, it may be better to frame certain math topics as “it’s easier/more difficult than it appears” rather than a flat easy/hard rating.

Much like with a musical instrument, there’s a mental resilience someone must have where they won’t be good for a while until they reach a certain point. Though someone may have the desire, they also must accept it will take time thinking about it and understanding it for math knowledge to set in.

The other difficulty is mathematics is less knowing facts and more curiosity, or desire, to learn and figure things out. It also means processing logical arguments to explain and prove properties within mathematics. You want peers and other people around to discuss math with, because it’s how you develop ideas and improve in it. There’s plenty of connections between literature and mathematics as well, despite them being two different fields. Knowing math is similar to like knowing another language where ideas and concepts come from earlier ideas and concepts and build upon them.

Math matters and it isn’t only for smart people. There’s no virtue in being bad at math, but there are certainly ways to make people hate math instead of respecting it. If you assume math is worthless and you may never use it in your life, you probably used math during the day before reading this.

As a warning to readers: do not assume any sections in this chapter are a substitute for attending a class or formally learning the material. That’s like opening up a winery and thinking the only job is selling wine (it’s definitely not!).

The Pedagogy of Math

In the United States, as of November 2025, math skills in students are not at the level they should be (University of California San Diego, 2025).

Though my primary subject matter is Science (and Engineering), Math is still necessary to properly understand those subjects and I find myself teaching Math alongside Science quite often. Science deals with many of the same issues, but across different and/or applied contexts.

  • Notice: I’m not saying other subjects are or are not suffering from similar issues.

The foundation of math in general is based on understanding and intuition.

Say for example I want to introduce you to a new proof. It requires abstraction, which means confusion and difficulty when you first see it; that’s normal. It also requires a foundation to work upon, because it may seem boring and useless at first and doesn’t get “fun” until you reach the later stages. Short-term memorization only takes you so far; you have to actually learn math to advance in it and do well.

It’s like the chicken and egg problem. You need students to buy into what you’re learning, which means showing math (and science) is useful and interesting. Those same students, however, may not see it that way until they’re actually good at it though! This is why practice (and homework, by extension) is still essential.

There’s also an issue with subjects where students may not care for abstraction and care more about the tools needed to solve their problems. This isn’t inherently bad, however, as every part of every subject cannot reasonably cater towards every student and their goals. Generally speaking, the wider the audience, the more a class may focus on application instead of theory/abstracts.

As for ways to get people to hate math, I can easily name a few:

  1. Doing math assignments as a punishment.
    • By extension, punishing a student every time they do it wrong.
  2. Student surrounded by a culture where being bad at math is cool.
  3. Students not accepting they’ll be bad at math at first before they get good.
  4. Students refusing to practice the math to establish a proper foundation.

Sometimes the explanation from a teacher is also part of the problem with math education. Say for example I present to you a proof. While explaining that proof, I also jump several levels of abstractions at a time or perform steps you may not rationalize yourself at that moment of time. The material I’m teaching may also not consider specific motivations or assume intuitions properly as well. It may also not include concrete examples with said abstractions to better connect the dots on how it works and why it works.

Clarity is important. For math, it is especially important, because a culture or group of people may not be interested in math, so it’ll be harder for them to learn as a result. You don’t need to, or should, sacrifice rigor to achieve clarity either; high rigor can complement high clarity. Definitions and formulae can be unambiguous and have strict justifications, as well as motivating examples.

You also should not completely sacrifice memorization for abstraction and conceptual thinking. Memorization is important for doing things quickly and accurately, like mental math, and ensuring it doesn’t take someone 1-2 minutes to solve 6x7 without a calculator or equivalent tool. It’s best for rules and axioms and memorization is on the level of memorizing the alphabet. You should know the alphabet, but you need further abstraction, concepts, and practice to interpret what each letter means, string together letters into words, then to put words into sentences to accomplish something.

  • To put it another way: you “memorize” systems, stories, and processes to solve problems.
  • Memorization is also a way to avoid looking up something every time you need it.

Despite the potential desire to explain proofs and reasoning to students, some students may not care and instead want the shortest sequence of steps. They’re mostly in it for a grade, rather than learning. If it’s a couple students out of a large classroom it’s not a major issue. If it’s most students though, you may be sacrificing productivity and deep learning to stick with high-level concepts instead.

Back to the Basics

Basics as in functions, decimals, percentages, fractions, times tables, and rudimentary algebra. This also includes operations with integers (basic integer work), mental math, subtract, divide, add, multiply, doing math without a calculator, the nature and use of a proof, and more. Many of these basics require memorization; there’s no way around it for how you’ll learn it and retain it throughout your life.

It also includes concepts like how an equals sign (=) means replaceable, the nature of numbers and number sense, order of operations (PEMDAS or GEMS) and what a percent actually means.

It’s also a touch concerning when I cover a topic and people don’t know division, fractions, decimals, and percents are related to each other, if not different ways to write the same thing. It’s even more concerning when students who are adults, or almost adults, cannot perform simple mental math and struggle with the basics like subtraction and addition.

Reading is important because… well, if you cannot read, then you’ll be unable to interpret even the symbology behind equations or apply the math you learn effectively. The same symbol can mean two different things across two different types of math, science, etc. Its why knowing and understanding context is extremely important. You cannot escape language arts that easily!

  • Students can solve many of their frustrations by simply reading the problem(s) too.

Basics also includes physical manipulatives for activities, like counting out coins and cash when people pay, because you can interact with them and they cross language barriers.

If I were to tell you what minimum level of mathematics a functional adult should go through sometime in their life or have an intuition of, whether through self-learning or formal education, I would say the following:

  • Basic Data Analysis
  • Calculus
    • Which also means Algebra and Trigonometry
  • Probability
  • Statistics

If I were to tell you what minimum level of mathematics every student should leave an education system with, because they’re likely to use it in “real life” and I’d want them prepared for when math is used against them, it’s the following:

  • Arithmetic (Addition, Subtraction, Multiplication, Division)
  • Mental Math and Number Sense (e.g. 7x7 or 6x21 solved quickly without a calculator)
  • Logic Rules (True/False and If/Then/Else)
  • Fractions, Percentages, and Ratios
    • e.g. a 1/3 pound hamburger has more meat than a 1/4 pound hamburger
  • Algebra (i.e. Abstraction)
  • Units and Dimensional Reasoning
    • e.g. 1 liter is bigger than 100 milliliters
  • Basic Probability and Estimation
    • e.g. a d10 consistently has a higher average than a d8, despite a larger range of numbers
  • Communication & Data Interpretation
    • e.g. correctly reading graphs of various types, like bar and line charts

“But I’ll Never Use This”

Alternatively: Why are we learning this?

If this question comes up, I suppose these responses or something similar suffices. It depends upon the group of students.

“I don’t expect you to use these skills in your everyday life. I do expect you to learn the things required to perform well though, like problem solving and critical thinking. That, and you need to pass this course for graduation.”

Or

“You might not, but your classmates will.”

Algebra and Trigonometry

These are important because they’re used practically in many ways seen and unseen. Examples include:

  • Excel and spreadsheets
  • Cooking
  • Making triangles
  • Percentages

Algebra also puts into writing many concepts you may intuitively do already, such as figuring out how much more money you need to afford X. You can do things without Algebra, but knowing it enables complexity, grants knowledge for practical tools, and grows thinking in abstracts.

Even “2+2=4” (or “a+b=c”) is technically Algebra when you introduce variables instead of numbers. There are many concepts and properties requiring learning Algebra as the basis to comprehend their existence. It generalizes mathematics and shows how numbers can transform through various operations.

It also introduces you to many other concepts you’ll see in advanced math, such as polynomials, simplification, and inequalities. Many people will stick with classical Algebra and not get into Abstract (or Modern) algebra, as the former suits many use cases in practice.

Geometry serves as further introductions into logic and proofs and reinforces critical thinking and spatial thinking. You can also utilize Algebra in Geometry as well. Without geometry, you’d be hard-pressed to interpret shapes, spaces, and many parts of a home you’re living in.

For many people outside of math-heavy fields, like design or art, you’re missing out on much potential without learning Geometry. For people in technical fields, like architecture and engineering, and not knowing Geometry means haphazardly creating a building plan that won’t pass legal codes and strict standards.

Calculus: Why it’s Important

Why Calculus? A lot of applied math loves using it.

One of the few times I’ll just say my primary source helping me here is from Paul Dawkins’ (Lamar University) class notes posted on the Internet. Hopefully I’m not held to trial for relying heavily on a single source.

  • These were available when I was in college going through the levels of Calculus back in early 2010’s.
  • There is a disclaimer, for Calculus I at least, that “you know Algebra and Trig[nometry] prior to reading the Calculus I notes” (Dawkins, 2023).
    • This is NOT a suggestion either. From personal experience, the hardest part about Calculus was the Algebra (and Trigonometry) and you’ll fail it without solid foundations in those areas.

I can, however, tell you once you understand higher level math (the what) and understand abstraction of math (the why), a lot of doors open for what you can do with it.

Calculus is a gateway to modern/advanced math and science, but is mainly important for two reasons:

  1. Rate of Change (Derivatives)
    • Δx (Delta (uppercase)) = quantifiable/finite difference of X between two values
    • dx = derivative, or differential of X, for instantaneous rates of change
    • ∂x (del) = partial derivative; the rate of change of X of one variable while other variables kept constant
    • δx (Delta (lowercase)) = small, but finite, change between d and Δ of X (e.g. deviation/error)
    • ∇ (Del/Nabla) = vector differential operator typically for gradients, divergence, and curl (rotation); often seen with partial derivatives
  2. Accumulation of Change (Integrals)
    • ∫ (Line integral) = Continuous accumulation of change in 1 dimension (dt) (e.g. line/edge of a cube)
    • ∫∫ (Double integral) = Continuous accumulation of change in 2 dimensions (dt²) (e.g. surface/face of a cube)
    • ∫∫∫ (Triple integral) = Continuous accumulation of change in 3 dimensions (dt³) (e.g. volume of a cube)

These are the most common (and “basic”) forms of integrals and derivatives I’ve encountered within Calculus when I went through it. How to solve them, their nuances, and associated theorems behind them are beyond the scope of this chapter, but exposure to concepts is still useful here.

  • You may also see closed line integrals (∮) and closed surface integrals (∯). They function similarly to line integrals and double integrals, except closed line integrals are over a closed curve and closed surface integrals are over a closed surface.

There’s two other notations which are similar to the above, but aren’t considered change.

  • Σ (Sigma) = Finite sum (addition) of discrete changes
    • E.g. sum(i=3 to 6) 3i -> (3x3) + (3x4) + (3x5) + (3x6) = 36
  • Π (Capital pi) = Finite product (multiplication) of discrete terms
    • E.g. product(i=3 to 6) 3i -> (3x3) x (3x4) x (3x5) x (3x6) = 29160

To more practically explain one form of derivative and integral relationships, I’ll draw from physics and use displacement (x), velocity (v), and acceleration (a). Each of these derivatives/integrals is measured with respect to (w.r.t.) time (dt).

  • Derivative of displacement w.r.t. time is velocity, which is rate of change of position over time.
    • v = dx/dt
  • Derivative of velocity w.r.t. time is acceleration, which is rate of change of velocity over time.
    • a = dv/dt
  • Integral of acceleration w.r.t. time is velocity, which is accumulated change in velocity over time.
    • v = ∫a * dt
  • Integral of velocity w.r.t. time is displacement, which is accumulated change in position over time.
    • x = ∫v * dt
  • Double integral of acceleration w.r.t. time is displacement, which is still accumulated change in position over time.
    • x = ∫∫a * dt²

In oversimplified terms, derivatives are like the slopes of a function and integrals are like the area under the curve on a graph.

Its applications are in areas you may expect, like engineering, physics, and architecture, and areas you would not expect, like retail. Here’s some further examples of Calculus applications below:

  • In physics: solve for velocity and acceleration, like in the example above, but also determine work and energy.
  • In engineering: optimize systems, calculate material stress, fluid dynamics, and simplifying complex systems into algebraic equations for applied engineering problems.
  • In finance and data science: costs, profits, trends, seasonality, and rate of production.
  • In medicine and biology: population dynamics, drug diffusion (concentration over time), and resource consumption rates.

If there’s any problem in any field involving rate of change or accumulation of change over time or space, Calculus is a tool potentially able to make a solution for it. Calculus remains an extremely powerful way to model, predict, and optimize.

As for the argument about not needing to know Calculus because you can solve problems with Algebra, there’s a reason why you can solve many problems with Algebra instead. The people designing those formulae and methods know the complicated ways and arcane magic behind the math (i.e. Calculus and beyond), so they can distill it into simpler, practical forms and prove their solution works to solve a given problem.

  • This means most people are using shortcuts because others already did the heavy lifting for them, but making or altering those shortcuts requires deeper knowledge.

This also isn’t a call to ignore math as a discipline or disrespect math either. The big things to also learn in Calculus is the reasoning behind all these theorems, creating useful abstractions, seeing the power of certain approaches to deal with various problems, and why it’s needed to develop more technical tools. Learning the inner workings and how to work with them gracefully serves you better over brute-forcing a single, correct way to (hopefully) solve a problem or throwing out a random word problem and calling it an application of math.

Beyond Calculus

I’ll admit: many people don’t need to go beyond basic Calculus. If we’re going off purely benefits vs time investment, you’re typically better off solidifying foundations and basics ten times over instead of building up a (potentially) precarious foundation.

It’s like how you can recall a topic and the general direction of what you need, then use a search engine like Google to find the details. The average person can stand upon the shoulders of giants and benefit from the collective knowledge of humanity throughout time.

In case you do get into higher math, the best way to describe it is it’s where numbers… disappear. It doesn’t mean numbers become non-existent; just more abstract and dependent on your notion of numbers.

That might sound really confusing at first and validly so. I will confirm, however, it doesn’t mean the math gets easier. It gets harder.

When previously you may’ve learned various formulae and how to apply it, and gotten away with just that, now you need to know proof(s). You have to argue why an equation, formulae, or theorem works the way it does. These proofs are not necessarily exclusive to higher math, and could be learned at an early age, but are definitely more abundant in higher math. Depending on how math was taught, someone may’ve experienced proofs far earlier in their learning journey.

Why proofs matter is because you need to convince people that something is true. A lot of math is taught by assuming that X is used to solve a situation, so X is the correct formula to do so. Unlike law or social arguments, however, there is no “reasonable doubt”; a statement can only be true or false and must be proven beyond reasonable doubt it followed the rules.

  • In other words, you’re learning more how you can solve things without numbers readily available and deterministic models unavailable.

You encounter more words with precise meanings, such as if, then, when, such, or, and, else, and so on. You deal with far more variables and symbols as placeholders, which is tantamount to death by a thousand cuts if you never got the foundations down. Sometimes things look like simple algebraic equations, but have a long and complicated proof steeped in higher-level math.

Additionally, what you could say in ordinary language, like English, now gets its own special notation through quantification and set theory! To give a small sampling:

  • ∀ (turned A) = “For all” or “for every”
  • ℕ = Set of natural numbers
  • ∈ = Belongs in a set, such as “is in,” “belongs to,” “element of,” or “ “is a member of”
  • { } (Braces or curly brackets) = Set, or collection of elements (e.g. {1,2,3})

With just that small sample, you can make something like this:

  • ∀x ∈ ℕ, x ∈ {1,2,3} ⇒ x ≤ 3

Which means “For every x in the set of natural numbers, if x belongs to set {1, 2, 3}, then x is less than or equal to 3.”

Half the work is just interpreting the equation. Things get weird and you really start delving into the unknown.

As a reminder, I’m not expecting the average reader to go above and beyond to the level where they, for example, seamlessly interpret the entirely of Kimi Linear. Knowing the high-level concepts will suffice for most cases.

What about Machine Learning?

Normally I’d place this under beyond Calculus, but it’s its own section due to popularity of AI in 2026.

You could learn the concepts about machine learning early on, even at a young age, provided you’re able to reason through it and read tables, graphs, and charts. For example, I could tell you about supervised vs unsupervised learning, structured vs unstructed data, streaming vs batching, and what even is machine learning without too much difficulty. The basics are doable.

The next stage is going into the high-level overview of concepts, like linear regression, gradient descent, hyperparameters, learning rate, and so on. I could give you a less technical explanation of these and you might go away with understanding it just fine.

The issue is when you need to go beyond high-level overviews and working with them in practice. At that point, the math provided in courses like Calculus, Linear Algebra, Optimization, and so on is no longer a suggestion but a requirement. Without that foundation, you’re going in blind, unsure why X is doing Y thing, and far more likely to break something or make things worse. You also cannot adapt its underlying concepts to the problems you need to solve as easily, which often means you’re stuck searching for a solution that barely works and takes a lot of time to get up in the first place.

If your goal is to get into and/or work with machine learning, you’d best learn math.

Bibliography

  1. Dawkins, P. (2023, October 9). Pauls Online Math Notes. Tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/

  2. Senate-Administration Workgroup on Admissions Final Report. (2025). University of California San Diego. https://senate.ucsd.edu/media/740347/sawg-report-on-admissions-review-docs.pdf

  3. Poincaré, H. (2011). Science and Method. Dover Publications.
    • Original Published in 1908.
  4. U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), various years, 1990–2024 Mathematics Assessments.
  5. Wikipedia contributors. (2025, December 17). Glossary of mathematical symbols. Wikipedia. https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols
  6. Zhang, Y., et al. (2025). Kimi Linear: An Expressive, Efficient Attention Architecture. ArXiv.org. https://arxiv.org/abs/2510.26692

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