Progressions in Math

(An Introduction to Your Future Math Classes)

An exceptional teacher is distinguished by a constellation of critical attributes. Among these include a mastery of the material, an innate desire to teach, a modicum of patience, a splash of pizzaz for entertainment, a creative approach to communicate concepts to various learning styles, an element of storytelling, and more.

I had a professor in the Acoustics Department at Pennsylvania State University who highlighted one of the biggest mistakes teachers make, precisely because he did not make it. Here’s what he did – and this is particularly important for math classes, but applicable to all subject-based learning.

He allocated the first few minutes of each class to provide the structure of what we would learn and why. In other words, he gave us the roadmap of where we were going (and where we had been).

His classes started like this,

“Recall the textbook chapter we are studying right now is titled X. We began this chapter by learning A, B and C. Last class, remember we learned D.”

He would then do a mini recap of D, what we discussed and learned during the last class.

Oh yeah. I remember now. Got it. I would think to myself and feel reoriented.

“Today, we are going to discuss E. It kind of looks like this and piggybacks off of A, B and D… not so much C. This is important, because we need to understand E to get to F, which is really the useful part of the chapter because F enables us to [fill in the blank].”

“OK. With that as the review and background, let’s start talking about E.”

And off he would go lecturing about E.

Maybe this just caters to my specific learning style, but somehow, I think this approach provides a universal benefit to students.

Here’s an analogy.

If you are biking around a new city with someone from that city, you’ll generally rely on them to navigate while you follow. However, it’s still nice of them to say (or signal), “We are going to hang a right at that next street,” before you actually get there.

If they don’t communicate this in advance and just suddenly turn right, you either miss the turn or you risk colliding with each other, or you hit the brakes, only to have to pedal harder to catch up. Either way, after a while of this, you start to feel a little like a small dog on a short lease, getting unexpectedly jerked around. Most people do not like that feeling.

Likewise, if teachers do not provide students with the roadmap of where they are going, or at least tell the students where they are going next, the students are mentally whipped around like the little dog on the leash. It’s also good to understand that you are in fact going somewhere. In the U.S. high school education system, this isn’t always entirely clear.

Without some navigational component, learning can be frustrating, even if the students don’t exactly recognize why.

Anyway, I picked up this teaching tip from a great professor in grad school and try to model it myself when I teach.

This made me think about the progression of math classes I took through my academic endeavors, from elementary school through graduate school in engineering. To my recollection, no one ever mapped out the full scope and plan of what was about to come BEFORE I started.

Apparently, the big picture is supposed to be a secret. Or perhaps no one bothered to lay it out nicely in advance.

When you attend math classes in high school and college, most teachers do not tell you the end destination and the path you will travel to get there. That is, what you expect to learn for the course. There’s no grand summary, no lay-of-the-land orientation to say, “We are starting here, and we will eventually end over there. But to get there, we will cover these terrains. Some of the parts along the way might seem useless (“When will I ever use this in real life?”) but we’ll need these intermediate tools to get to the final destination, which will be useful.”

Instead, math classes often start with, “Open your textbook to chapter 1.”

FAIL.

I’m continually amazed that the first few days of any class (especially math) are not dedicated to summarizing the whole semester.

It’s like taking the kids on a long trip and not telling them where they are going and not giving them a map of the expected route and the surroundings.

So, this is that map… at least for math.

[This piece probably benefits a high school or college age audience. I’m pretty sure there are very few people in that demographic that read my blog. But I don’t care. This was what I felt like writing about.]

Here we go…

Basic Math

The fundamentals of math boil down to the basic operations one can perform on numbers and equations. For such a complex topic, there are surprisingly few fundamental operations. Well, it’s a lot to learn, especially if you intend to fully understand them, but it’s not like there are thousands and thousands of them.

Basic operations are the primary colors of math:

• Adding
• Subtracting
• Multiplication
• Division

…in that order. Well, technically, that’s the order you learn them. But you perform them in exactly the opposite order when solving math problems.

These basic operations are followed by secondary operations. These are the greens, oranges, and purples of math.

• Fractions
• Decimals
• Exponents
• Logarithms

These secondary math operations are mostly just efficient ways to denote the basic operations. They are shorthand for the four basic operations.

Fractions and decimals are just another way of representing division.

Exponents and Logarithms are a more efficient way of writing multiplication (or division if the exponent is negative).

In learning these secondary operations, we haven’t really learned anything new, fundamentally, just more clever and efficient ways of denoting the things we already understood. Nevertheless, understanding how to write things efficiently (most simplistic form) is a significant part of learning math. If we do not learn how to write and understand simplified syntax, we would have a very difficult time approaching more complex math problems.

After we master the basic and secondary operations, we move on to math courses that now have names that are not simply “Math”. In the U.S., math courses are called “Math” until about the 8^th or 9^th grade when students start Algebra I.

This is the order of math courses you will likely take after just “math”:

• Algebra I
• Geometry
• Algebra II
• Trigonometry
• Calculus I
• Calculus II
• Calculus III
• Differential Equations
• Partial Differential Equations
• Linear Algebra
• Imaginary Number Theory

Let’s have a look at each of these in turn, just for fun!

Algebra

Math pretty much makes sense up to 8^th grade when you hit algebra. Prior to that, it was basically versions of the fundamentals (adding/subtracting/multiplying/dividing) and the secondary operations (fractions/decimals/exponents/logarithms). And the dreaded “word problems”.

Prior to algebra, the main challenge was likely “long division” and fractions. But generally, we understood “long division” to mean “division, but longer”, a longer version of the same thing. That’s the good part of the name “long division”. It’s quite descriptive.

Even if you didn’t quite grasp it in third grade, there were more years of practice to follow. And, it’s a fairly procedural task. Once you learn the steps, just follow the procedure and the answer eventually pops out, magically on top of the long division line, plus a remainder, sometimes.

Once you hit algebra, they introduce letters into the equations. At first, you think, “Hold up. You can’t add or subtract letters.”

It takes some time, but eventually, you learn,

In algebra, letters are just substitutes for numbers, when you don’t know the number yet.

Generally speaking, the goal of algebra is to determine the actual number the letter represents.

They make it seem considerably more complicated than that, as if there’s some mystery to remain always hidden from you about this part of math. But, the bottom line is, find X. Once you know X, the problem is solved or generally solves itself from there.

Where they try to trick you is making you decide what X should be. The word problems just got a little more complicated. But, with time and practice, you get the hang of it.

In algebra, you also learn how to figure Areas for various shapes (like squares, rectangles, parallelograms, rhomboids) and Volumes for 3-D shapes (like boxes, spheres, prisms, etc.)

Geometry

Just when you thought you were getting the hang of it, including the letters, geometry comes along and throws you a complete curve ball.

Now, from the name, we might think geometry is about shapes. That’s kind of what the name implies. And, to a certain extent, I suppose that’s what geometry is about, but, that’s not what you spend your time doing when you have geometry homework.

Geometry is about proofs.

Starting with the most basic assumptions, you work your way up in complexity to “prove” something is true, based on your previously established truths/proofs.

Proofs are laborious.

As far as I was concerned, I could never be quite certain that my proof was true, because no matter how many steps I inserted into my proofs, my teacher (and the answer key in the book) always had additional intermediate steps that I thought were self-evident. Apparently, you can’t leave out those bits and have a solid, watertight proof.

The great thing about algebra is that you often do a certain number of predefined steps to solve the problem. With geometry, you already know the solution, the answer (the thing you are trying to prove), you just don’t know how many steps you’re supposed to use to get there. Good luck.

Of all the math classes I took, geometry was my least favorite. Sorry kids. Suffer through it.

Recently, I was at a neighbors’ house-warming party. Their daughter mentioned that geometry was one of her favorite classes, a sentence and sentiment never before uttered by a student since Euclid wrote his 13-book treatise on geometry (~300 BC) that set forth the basic axiomatic principles like “Any two points can be joined by a straight line”. To the 9^th grade me, that’s an example of something so basic, it didn’t need to be said and certainly didn’t need a formal proof. But you can’t disagree that geometry is thorough.

I’m sure geometric principles are useful if you aim to be a builder or an architect. But otherwise, it seems to be the one math class with the least utility for any job you might ever have. The only real argument I have for surviving proofs is that it helps develop your ability to think logically, a super useful skill in any endeavor because it shapes your brain to build logical structured thoughts, grounded in axiomatic principles.

So, suffer through it. Retain the ability to think logically to determine truth (as best you can) and to arrive at logical conclusions to solve real world problems, but forget all the basic proofs you will establish in this class. You will never (and I mean NEVER) need this again in your life.

Algebra II

Finally, back to something you recognize from before, but largely forgot because you were forced into a year-long sabbatical from algebra trying to wrap your brain around geometry, to no avail.

Algebra II is back to the familiar. Fortunately.

Algebra II is all about the quadratic equation, the Pythagorean theorem, and the standard form for the equations of basic shapes like lines, parabolas, circles, ellipses, and hyperbolas.

In that order, because that is the order of complexity.

On the surface, that might not seem terribly useful, in that you already know how to draw a line before algebra. But a line describes the linear relationship between two variables. This can be useful as a first order predictor of what might happen next, given certain initial conditions. Or what happened before, given the end result we now know and the process something likely went through to get there.

An equation for a line is a predictor of sorts. The simplest one, which is why we start there. But, because most things are not related linearly, we quickly move to higher order relationships and add either terms with exponents or we add more terms to the equation… or both.

While many things in life are considerably more complicated, and therefore non-linear, some things truly are linear, like driving at a constant velocity down the road. You go X distance per unit time. It’s linear.

Springs are also linear in that the force is directly proportional to the stretch or compression unless you go too far and over-stretch the spring. We’ve all done that in our youth or had a friend do it to our toy. All Slinkys eventually get abused in this way. It doesn’t go back after that, and pretty much ceases to be a functional spring… but before that, it’s a simple linear equation.

Armed with an equation for a line, and letters for variables, Algebra II starts to solve some real-world problems. Or, at least approximates solutions.

Except for lines, all the other equations mentioned above are second-order equations (meaning one of the terms is squared). These can be more useful than just a line because we now have second-order predictors of what might happen or what did happen… just like we did for a line, but now with greater accuracy.

We’ll throw exponents in the mix for Algebra II, as an added bonus. But exponents are not difficult, so you won’t spend a lot of time on this.

That said, the opposite of exponents, log functions, are quite difficult to get your head around. For some reason, log functions are not as intuitive.

Truth – no one really understands logarithms the first time seeing them. Fortunately, the process by which we use them and solve for them is quite simple and formulaic (no pun intended).

Log functions – easy to solve, difficult to understand (at first). But it’s graspable and you’ll get it with time and practice, like most things.

Trigonometry

As if geometry wasn’t enough of a trip, trig is quite different from anything you have seen before in math.

Trig is the one math subject where it seems either you get it, or you don’t.

Fortunately for me, trig came easily, mostly thanks to Mrs. Johnson, a stellar teacher at my high school. Of course, a great teacher makes all the difference.

Unlike geometry, trig really is about shapes.

In trig, you’ll learn new ways to calculate angles and lengths of the sides of shapes (often in radians, instead of degrees), using trig functions.

The basic trig functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

All the other trig functions are deviations and derivations of these basic three. If you solidly understand these three base functions, the rest fall into place rather neatly thereafter.

You also learn a lot of different ways in which these trig functions might be combined such that they form complex relationships with each other, relative to certain shapes, particularly triangles.

Calculus I

Calculus exists to throw you for another math tailspin. “Calc” is a whole different sort of thinking around math. Again, different from any math you have seen before.

Algebra was mostly numbers with a few letters thrown in. Calculus is the first math class where you ask, “Where are the numbers? And why are we using Greek letters?”

Calculus is where you will learn most of the Greek alphabet.

Calculus is initially about limits, then derivatives and, later, the opposite, integrals.

To me, calculus is a fantastic class and the first time you start to think you might have approached “real proper math”… in that it’s useful in the real world. And interesting. And approachable.

Beginning from the first week with understanding limits, calculus is almost more a way of thinking. It’s a beautiful thing in which to train one’s mind in how to think.

Calculus II

Finally, a back-to-back math class where the rules don’t change on you too much. Even the name stays the same.

Calculus II is a natural continuation of Calculus I.

Calculus II centers around the concept of integrations, double integrations, and triple integrations.

Calc II explores how to calculate areas and volumes swept out by rotating shapes through space.

Calculus III

We learn in Calculus III, that there’s really only “Calculus”. The false division into three parts is simply because it takes that long and that much practice to absorb that much calculus, because you are also taking other classes at the same time.

Calculus III focuses on applying calculus, but as functions of multiple variables and vector functions. It also delves into polar coordinates (as opposed to the more traditional Cartesian coordinates we all know). Polar coordinates are more suitable for certain types of problems. Sometimes, we can significantly reduce the complexity of a math problem by reframing the coordinate system we use to solve it.¹

Strangely, I have fewer memories of Calculus III compared to the other math classes outlined here.

Differential Equations

As with most math problems, with Diff EQ, you are still trying to find X. It’s just that the starting point for the problem isn’t just “X”, but a derivative of X (dx/dy).

It turns out, Diff EQ is really about ordinary differential equations, which is a special case of differential equations, the easiest case, because they don’t contain any partial differentials. That’s the next class (see below).

Partial differential equations are more difficult to solve. Consequently, we start with the base case and work our way up in complexity, over time. This is much like finding the area of a square or circle before you learn how to find the area of a rectangle or ellipse. A square is also a rectangle, just a special case, where all sides are equal. A circle is also an ellipse, a perfect one.

This is much like algebra when you learn the Pythagorean theorem for right-triangles:

a² + b² = c²

You learn this first because it’s the easiest scenario and yet still useful to know, but it’s really a special case, because it only applies to right-triangles. It’s not until you get to trigonometry that you learn the more general relationship between the sides of ANY triangle:

a² + b² + 2abcos(C)= c²

[This relationship is called the Law of Cosines].

Well, that’s more complex and therefore not a good place to start, because you haven’t had trig yet when you are in algebra. (cos) is a trig function.²

The same thing is true for differential equations. You start with the special case, because it offers a simple introduction to the concept of differential equations. You then work up to more complex differential equations when you move to the next class, Partial Differential Equations.

The famous physics equation presented in Newton’s second law of motion, F=ma (force = mass times acceleration), is a differential equation… because acceleration is actually a “derived unit” – a derivative.

Acceleration is the change in velocity with respect to time (a=dv/dt). Further, velocity is the change in position of something with respect to time (v=dx/dt). Acceleration is therefore the second derivative of the position of something with respect to time (a=d²x/dt²).

I use this example to show that an equation we know well, is actually a simple differential equation, in that we don’t initially know X (the position of something), we only know it’s acceleration.

The point here is that Diff EQ is an introductory class to Partial Differential Equations. Just when you thought you were reaching the heights of math, you’re still really only taking “Introductory” courses.

Partial Differential Equations

Partial Differential Equations are more common in the real world, mostly because we live in three-dimensional space, plus time. These dimensions add partial differentials to our equations.

Partial Diff EQ’s are equations where the variable we are trying to solve for is not a number (like X), but a function of at least two other variables. So that starts to get a little more complicated.

The solution to a partial differential equation (PDE) is a function that solves the equation.

How do you know what that is? Here’s the tricky part… you kind of guess at it and work backwards. I wish someone had told me that when I started this class.

PDE’s is like calculus, but backwards. It’s like starting with the answer and trying to figure out the question.

PDE’s would be really difficult to solve, except some geniuses figured out statements like this:

IF the solution [fits this general pattern], then the original problem must look [something like this general form].

Solving partial differential equations is therefore just a matter of seeing enough problem sets such that you can recognize and classify the various forms (because you’ve seen them before) and therefore have a good guess at what the form of the solution must look like.³

So, you start with a potential solution and work your way backwards to fill in the details.

It’s still a lot of work and solving a single problem of a partial differential equation almost certainly fills up 2-3 pages of paper (if you write small… and you will) but understanding the general categories of the problems seems to be the key for this class.

I’m not comfortable that I could still do PDEs without re-reading and re-learning the material. It would require a substantial review for me to come back to this level of math. It’s been too long since I’ve seen it (now that I’m 53!).

Linear Algebra

With a title like “Linear Algebra”, you might be lulled into thinking it’s back to algebra and lines. Yay! That’s what I thought. That’s what the title implies. But, no! That’s definitely not the case.

Linear algebra is almost pure matrix theory.

Matrix theory is an incredibly powerful tool set enabling you to solve systems of equations that would otherwise completely lose you in complexity, left to solve them by traditional methods (with pencil and paper). In fact, I’m pretty sure spreadsheets (like MS Excel) use matrix theory to solve things quickly (this is just a hunch. I could be wrong on this point).

Matrix theory is mostly about manipulating the values in the matrix such that you get a string of zeros along the diagonal. Somehow, this magically solves all the equations simultaneously. Very few people actually understand why this works. When you take Linear Algebra as a class, it feels much like voodoo. Voodoo that works. Good enough. Do the homework. Get the answer. Be amazed, but you likely won’t really understand why it works… just that it does.

I used matrix theory to solve a particularly difficult engineering problem during my second internship at an underwater robotics company in Houston.⁴ Something about using the theory in practice made it all come together and I understood matrix theory much better after that. And, like all math, the desire to learn it increases when you come across an actual real-world problem where it’s useful.⁵

Imaginary Number Theory

“Imaginary Number” is the single worst name coined for a mathematical concept. The name implies what you are studying is fairy dust. Imaginary. Disney World.

Given the name, your instinct might be to wonder why you should be compelled to learn something that only lives in the world of unicorns and dragons. Resist this thought because this is not the case at all.

Imaginary numbers play a real role in real-life math problems, especially electrical engineering and acoustics. There’s nothing imaginary about them. It’s just unfortunate naming really.

Once you get your head around the idea that “i” is just the square root of negative one and that it is NOT imaginary, then this class of math gets a lot easier.

Summary

Sometimes it’s difficult to imagine real-life applications for much of the math we learn in the various math classes we take. This isn’t because there are too few real-world examples, it’s because our teachers often fail to show us real-life context, use-cases where the math is applicable. Maybe some teachers haven’t used the math they are teaching in the real world.

Perhaps it would be a good exercise to invite industry practitioners to present real-world problems they have solved using the math the student is currently studying.

All math begs the student to find a practical application such that we might desire to learn it… precisely because it’s useful.

Rare is the student who thoroughly enjoys learning something impractical for the sake of exploring the depths of it to no practical purpose. Life is too short for this.

This is why students frequently ask, “When will I ever use this in the real world?”

Of course, underpinning this question is the idea that the world is in fact, real, and not imaginary. 😊

Bonus – Statistics (and some ranting)

Is statistics really math?

It involves math, but I’m not entirely sure it’s a math class, in the purest sense.

Statistics, in its most practical form, aims to calculate the probability of something occurring and to make a statement about the uncertainty of your answer/guess.

You don’t have to think very hard to understand why this has ample real-world applications.

The main problem of statistics is the misuse of the output it provides.

Just because you can calculate something, doesn’t mean the resulting answer is grounded in truth. This is normally due to these factors:

1. The input assumptions were incorrect from the start.
2. We didn’t include all the external factors that might influence the event (primarily outliers and rare events).
3. We did the wrong type of statistics for a given problem. Specifically, we over-simplified the problem (usually assuming normal distributions with a non-gaussian data set).
4. We ignore or miscalculate the uncertainty of the answer.
5. We misinterpret or overly interpret the result, extrapolating it beyond what it actually says or implies.

Just like the math classes previous to this one, statistics teaches you the simple cases first and broadens out to more complex statistics as we layer on broader parameters (relax the assumptions to be more real-life).

The starting point in statistics is to assume things fall into a normal distribution (also called a Gaussian distribution… the famous bell-shaped curve). Sometimes they do. Often, they do not. But this does not seem to prevent people from using the basic level statistical analysis improperly.

The most common error in statistics is to apply statistical mathematical algorithms intended for normally distributed data when that data is non-Gaussian. This results in forecasting probable outcomes that are entirely incorrect. The most common result is an understatement of risk or error rates.

I have personally seen this many times at work, both in engineering work and especially in finance, where people have been armed with basic statistical knowledge and apply these elementary equations to non-Gaussian data. Almost all financial data is non-Gaussian… in that it does not fall into a normal distribution. With money and finance, the curve isn’t as pretty and symmetrical as we might want it to be, even if all the data we have show it to be normally distributed.

Just because you haven’t seen the outliers in your sample data doesn’t mean they won’t ever surface. It just means you haven’t recorded it yet.

I’ve also seen this error in weather and insurance applications. Take the “100-year flood plain” as an example. Generally speaking, insurance companies won’t insure your house for water damage if you build it inside the 100-year flood plain. Makes sense on the surface.

This insurance rule exists in most states, including in Oklahoma, where I grew up. But how long do you think people have been measuring the flood levels in Oklahoma? It wasn’t even a state until 1907. That’s barely 100 years total… and I’m pretty sure the original settlers weren’t going around recording this type of data in between harvesting their crops and wrangling the cattle.

But, for the sake of argument, let’s say they have been collecting flood-level data for 100 years. What if a massive flood comes along in year 101, just one year later, and exceeds all previous levels? Oops. That just wasn’t in the data. Unpredictable.

Further, suppose we have built concrete and asphalt structures, dams, lakes and ponds, and have therefore changed the dynamics of water absorption and run-offs patterns through tera forming (as we have). Surely this also affects the 100-flood plain. Certainly, we do not have 100 years of data for the built-out landscape we now have.

Several hundreds of years of data are required to estimate the probability of a flood at certain levels within a hundred-year span.

But now I’m off-topic and no longer talking about progression through math. The point being, of all the math classes, statistics is the one that might be the most useful and simultaneously most dangerous and potentially most harmful because it is misused (and misinterpreted) so frequently.

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FOOTNOTES:

1. This is also a life hack.
2. Note for Nerds: for a right triangle, the angle C is 90 degrees. The cos(90) is zero… so this third term goes away in the special case of a right triangle. That’s why we start with the basic Pythagorean theorem for right-triangles.
3. This is also a good life skill.
4. I was given this challenging assignment precisely because the senior engineers working there had not seen this type of math since they were in school and had long forgotten how to solve it. “Give it to the intern. He just took the class.”
5. Perhaps this is what is missing from a lot of math classes… showing the real-world practicality. This notable gap hinders students from learning because the application is disassociated from all the rote equations.