Math Misery? I Think I Know Why

On any given day when someone finds out that I am a mathematician, I am then regaled with tales of woe. The stories are the typical one- or two-liners that go something like this:

  • I liked math until the letters came in.
  • I used to be good at math until I took Calculus.
  • Fractions killed me.
  • There were so many formulas to memorize.
  • My \(nth\) grade teacher sucked.
  • I liked Geometry but couldn’t do any of the proofs.
  • I was never good at math.

That was the not-completely-jaded crowd, but certainly a set of people who felt that they had math inability.

Then there’s the “I hate math crowd”. With that crowd, the sentiment is fairly universal and singular. “I hate math.” The reasons are many but all virtually the same — it was impossible, it was confusing, it was too precise, etc. The visceral reaction to anything with a hint of mathematics stinks of trauma. It goes beyond just a phobia, it seems to be much worse, I feel. (For the record, I am not trained as a psychologist, so I use words like “phobia” and “trauma” in a non-clincial sense.)

When finding out that I was a mathematician, a history professor said to me, holding his two index fingers up to form a cross, “You stay away from me! I have nothing but miserable memories about you, your kind, and your profession.” Eek. True story. What in the heck did I ever do to him?

What’s also interesting is that everyone with some form of math anxiety can state “how much math” they got to. It’s bizarre, if we think about it. It’s almost as if we’re talking about an old-school side-scroller video game. Ahhh, I died on world 3-1. And let’s not even talk about warping with insufficient lives.

I believe everyone’s story. I believe they had a terrible teacher. I believe they failed in Algebra or Geometry or Precalculus or even with fraction arithmetic. I’ve heard of grade school teachers say that they wouldn’t want to teach fourth grade because of the math. It’s fourth grade math! So then we wonder why we have shows like “Are you smarter than a \(\pi^{2}\) old?”

I think there are many reasons why people have the reaction they have to mathematics. What’s interesting though is that mathematics isn’t the only subject with the “I can’t …” mentality. I recently took up drawing and when I sit in a coffee shop learning to draw, I usually have a few people stop by and inquire about what I’m drawing. The same thing happens with when I am going through some math text. Curiously enough, the reaction I get from most people is approximately, “Oh I can’t do that”. They can’t draw. They can’t do math. That’s fine. I can’t do handstands. We all can’t do certain things.

With drawing, I don’t necessarily get the same hate- or fear-filled response. But a simple, “Oh I could never draw.” So the inability exists with drawing, but often not the hatred? Why?

Over the course of a semester, as I get to know my students, they start sharing their academic woes. “I hate Chemistry.”, “I hate Biology.”, “I hate English”, etc. But there’s a difference in the way they say they hate or in their general distaste for mathematics.

People will also say that they hate a certain sport in the context of what sport to watch (and I suppose that would mean also what sport to play). Golf is usually on the top of the list, followed by baseball and tennis dueling it out for second place. But still! It’s not the same hate as it is with mathematics.

I want to explain why an inability with drawing doesn’t have “hate” associated with it. Also, I want to explain why there are differences in the “hate” when discussing general science courses versus general math courses.

Why The Hate? Why The Pain? Why The Misery?

Let’s suppose you declared your college major to be Biology. You take Bio 101 and hate it. What do you do? Change your major! But guess what? The math requirement for that new major will still be there. There’s no escape from it. No matter what major one chooses there is some basic math requirement. Biology majors often have to take math coursework up to Calculus I (there’s the whole “up to” video game speak, again).

Want to change your major to an Engineering discipline? Then you’re taking courses well beyond Calculus I. What about Chemistry? Tons of math. Physics? At some point it practically is Math. So there go the S & E from STEM. How about computer science? Well, there you’ve got your Algorithm Analysis course, among other things. There’s lots of Graph Theory as well not to mention all the craziness with floating point arithmetic. And this is just the tip of the iceberg. With computer science out, there goes most of the T in STEM. And the M was already a non-starter.

What of Accounting? Accounting is certainly math heavy even if it happens to be a lot of arithmetic (which is not the only thing it’s heavy in).

And Finance? Good luck. Finance nowadays has been invaded by mathematicians, physicists, engineers, and the like. Any sophisticated work in Finance is going to be on par with the math found in Engineering majors. And even if we are not talking about PhD quant level math, the entry level position as an Analyst in Investment Banking is going to be toiling through spreadsheets filled with … numbers. A mainstay equation in Finance is about compounded interest: $$P(t) = P_{0}\Big(1 + \frac{r}{n}\Big)^{nt}$$

No escape there.

Thinking about Medicine? Then you’re absolutely taking math courses with the dreaded Statistics course rearing its ugly head.

The same goes for Psychology.

So, goodbye to STEM. This leaves us with the Arts and Humanities.

As far as I know, an overwhelming majority of Bachelor’s degree programs in the US require students to take at minimum an Algebra course, typically named “College Algebra”. But this is Algebra! In college! It’s almost baffling! Students have been taking Algebra since probably the seventh or eighth grade! Why are they still taking it? It’s because they never learned it in the first place and have just been shuffled up through the system. Colleges, however, don’t really bend on their requirements. If the student doesn’t place out of Algebra, then they are taking the course. And so it comes down to either learning Algebra and passing it, or hoping to get a professor who is somewhat lenient and will continue to let the student move through the system.

This is my point. There’s no escape. It’s a type of torture.

Notice, however, that if you took Biology 101 and hated it, then changed your major to say, Electrical Engineering, you will have still have satisfied a basic science requirement for that major and won’t need to take Biology ever again (unless you choose to go into some cross-disciplinary major like BioElectrical Engineering (if that exists, but you get my point)). Regardless, your hate for Biology starts and stops at Biology 101.

Or in my case, I never wanted to take Biology and had declared myself as an Electrical Engineering major. My basic science requirement was a bunch of Physics courses and either Biology 101 or Chemistry 101. I chose Chemistry 101 and never had to “suffer” through Biology.

So this is a main reason for the difference in hate. I can hate Biology (I don’t actually hate Biology) and never be bothered by it. But if I hate Math, too bad. It’s sitting around everywhere. Even if I earned a non-STEM degree and went out in the working world, basic math is always there. For the student that never got a handle on the times tables, integer or fraction arithmetic, they’re in trouble. They’ll deny it. They’ll say that they’ve never needed to use any of that math that they took and they’re doing just fine. They’ll identify arbitrary topics and specific facts and say, “See, I’ve never used that!”. But this is just confusing the issue with the broader subject matter. Maybe they are doing fine, but could they be doing better? Math inability is a silent killer.

If I were illiterate, we would all agree that that is a serious hindrance in terms of a career and perhaps other things. But not knowing the math that is covered in a basic skills course up to Algebra? It becomes harder to argue that there is a hindrance in not knowing this. This lack of knowledge is not immediately catastrophic as being illiterate. But this lack of math knowledge is a financial death by many many lost pennies, slowly, steadily over time compounding into large sums of forgone savings.

The narrative problem we have here is that that kind of math isn’t considered math, it’s called finance. Too bad. Underneath it all is mathematics, more in some disciplines, less in others.

In any case, no matter how much a person doesn’t want to take a math course, no matter how much a person would rather be ignorant to the entire subject matter, they have to take math courses, often at least up to Algebra (stupid levels). The ridiculousness of it all is that despite being forced to take the course and over so many years of exposure, people come out still not knowing the material, which means that they’re not going to use any of the knowledge they could have gained.

So why are people being forced to take this coursework? I have a lot of reasons for why, but that’s not the point of this discussion right now. However, to not leave you hanging, here’s a short reason: mathematics is easier to test on a standardized level than other subjects and mathematics is a vital part of practically all STEM fields — the set of disciplines that have allowed us our conveniences. Because of the testability of mathematics and its near ubiquity, it then becomes an easy filter: STEM or not-STEM.

The point is that as there is no escape from a certain minimum level of mathematics, what results is an emotional reaction. At some point it really does feel like torture.

I have yet to meet a person who says that they hate drawing. And why? It’s not a subject they are forced to take. There’s drawing and art education that happens at the elementary and middle school level. But once students enter high school, art becomes more an elective than anything else. Most people can’t draw well. We can all scribble around and draw stick figures and awkwardly shaped houses, but few of us are actually good at drawing. However, there’s no shame in the inability. Art isn’t a filter for anything unless if one wants to go into the arts.

So there’s one reason for why the hate is different for mathematics wholly versus other disciplines — math is just forced onto people for over a decade with the majority of students’ inabilities never being cared for.

It’s a cruelty and contradiction in our education system. If we think about it, we can see that our education system’s main purpose is not to educate. That’s a side effect. Our education system’s main purpose is to filter. If our education system’s main purpose were to educate, we wouldn’t have this systemic phobia and hatred of mathematics, nor would we have droves of students taking basic math courses after a decade of exposure to the material.

What I see with college students is defeat with their math class before it has even started and they have been defeated at an early age. They are just hoping beyond hope that the math course they are taking with me will be the last one. They are hoping they’ll just pass it. Eventually it has to happen right? They’ll have remembered enough arbitrary facts to gather enough arbitrary points to pass.

The first day of lecture is often filled with post-lecture questions like “I’ve taken this course three times, I just have to pass. What do I have to do?”, “Will this course be hard?”, “I’m not good at math, will there be extra credit?” None of these are the right questions. It’s all screwed up. They’re just hoping they’ll be pushed through one more step.

Math Misery Explained

Now, the above discussion doesn’t actually explain a root cause for math misery. The above discussion is explaining the symptoms that are part of the downward spiral into math misery. I want to get at something a bit more atomic.

There are lots of reasons that people (myself included) give for what’s wrong with math education — emphasis on memorization, teaching mathematics in a rote and mechanical manner devoid of meaning, silly tricks and gimmicks, etc. All of these are certainly poor ways to teach Mathematics. But there is one thing that people do not focus on whatsoever and is not part of the louder chorus of what bad teaching is.

Consider the following paragraphs. They’re the opening lines to a story I’ve had dancing around in my mind for a while.

I went for my usual midday walk into the forest, ironically, to clear my mind. I always like to walk alongside the creek listening to the water softly pass by. On this particularly normal day, I saw something unusual wedged between a fallen branch and a rock in the creek. It was cut with machine precision and had that familiar gray color with unfamiliar markings. I knew instantly that this was Zahl technology. But here? We haven’t seen the Zahl in ages.

Curious.

As I went to pick it up, I noticed that the creek was reversing course! The temperature had dropped suddenly and there was a great shadow that was cast over me. The hairs on my everywhere stood on end. I knew what that meant. It was a Zahl!

In a low, but booming voice, I heard, “That’s ours.” I dared not look back. I grabbed the object and ran. But no more than two steps into my run, it was magicked out of my hand. They had taken it. I didn’t care, I had to get out as quickly as I could. I heard the click of one of their heinous weapons. We may not have seen the Zahl, but we knew what their weapons sounded like.

Ok, two sets of questions:

  1. What did you think? Do you like it? Or is it a tired, hackneyed, predictable introduction to some crappy, predictable sci-fi plot? I’m still sketching it out. :)
  2. How did you read it? Did you skim around? Did you read it word for word?

Now let’s try this:
$$P(X \leq x) = \int_{-\infty}^{x}p(t)\ dt$$

Does this equation mean anything to you? If you’re a mathematician or have had enough probability theory, you’ll probably immediately know what it means, even without the appropriate and necessary context of the constraints on \(p(t)\).

If you’re not familiar with mathematics or at least with probability theory, the equation means nothing to you and my explaining its meaning isn’t going to help either. So I won’t bother. As far as we’re concerned it’s gobble-dee-goo.

Now let’s try something a bit simpler (hopefully):
$$x + 3 = 7$$

Most of us, whether forced or not, have had Algebra and can recognize the symbolism. We can perhaps even predict the question: “solve for \(x\)”.

But here’s the real question: how did you process \(x + 3 = 7\)? How did you read it?

This is one of the most atomic reasons for math misery — students are never actually taught to read mathematics. I’m not talking about word problems. I’m talking about the symbolism.

We’ve become accustomed to reading in chunks. The opening lines for the terrible sci-fi story I want to write don’t all have to be read. And within each sentence, we don’t have to read all the words either. We get a general sense of what’s going on. We can ignore the bad grammar. We can ignore poor punctuation.

As kids become adults and as they read more, they can develop the ability to read, more quickly. The same holds true with reading through mathematical symbolism. But there’s one difference.

When we read passages of text we have a feel for how long it will take us to read passages of certain lengths. When we see a 600 page book we can judge if we’ll have enough time to dedicate for it. When we see a blog post like this, which is now nearing 3000 words, we know approximately how it will take.

But!

Math misery sets in when we see an equation, something as “simple” as \(x + 3 = 7\) and we don’t immediately understand what it says. It’s five symbols! Here’s a five-letter word: “maths”, the way every other nation shortens mathematics.

Our expectation for how long it should take to process \(x + 3 = 7\) is wrong! We’re using our intuitive approximations for how long it would take to read text and subconsciously applying it to the math text. Then we get frustrated that it doesn’t make sense in the half second that we took to read it. We throw our pencil down and declare that we hate math.

When I sit down to “read math”, I make a mental adjustment. I tell myself that I’m not reading a storybook. I tell myself that equations that look unfamiliar need to be thought of as a page of text and should probably take that long, if not longer to read. I also tell myself that I’m reading a legal document. Any time I read through legal documents, I read through them as carefully as I can. Those documents are precise, as is with math text. We can’t just ignore symbols.

It is a simple mental adjustment for students to make and for teachers to elaborate on. Math misery sets in when students don’t make this mental adjustment. If the subject topic is new, then just pretend that every equation is the equivalent of reading 1-2 pages of a storybook and to give it that time. Eventually and with enough familiarity, reading something like $$\sum_{k=0}^{\infty}\frac{1}{k!}$$ isn’t laborious and can just be read over as if it were part of a story.


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