by Nicholas Homberg

June 2022

Math is hard for humans to fathom. We were the progenitors of mathematics and yet it’s still a struggle. But, why? First, it’s an abstract subject and abstractness is fundamentally difficult to grasp. Second, our brains are wired to “do what we enjoy” and it prefers convenience; math is often unbefitting of either. Lastly, school could have an ironic hand in this issue. Let’s discuss these topics in order to best address the question: why is math hard?
**First, math is abstract. **

In the article “Is Math An Absurd Subject?” by Manpreet Singh, they explain that, “Mathematics deals with numbers – and numbers are not real elements. They are fictitious theories… in reality, we cannot experience numbers… Nobody has ever found a five in the wild. We can only find five of something— such as five elephants, five apples, five eggs, and so on… Mathematically speaking, the term abstraction is a cognitive representation of objects, where the student recognizes topics to construct a mental image of its own, known as abstract”.

Essentially saying that math fundamentally doesn’t exist, it’s immaterial, and only by it describing a thing can it be understood; and that’s how it’s abstract. But, why would it being abstract make it difficult to understand? After all, anyone who can use a metaphor can think abstractly, right?

Well, in the article, The Difference Between Concrete Vs. Abstract Thinking by Julia Thomas (medically reviewed by Tiffany Howard, LPC, LCADC), she outlines that while concrete (which is thinking about the literal or tangible) and abstract thinking are “the two main types of thinking,” there are actually several other types: Creative, Critical, Analytical, Divergent, Convergent, Linear, and Nonlinear.

“The average person who is already largely an abstract thinker could learn how to become even more so. One way to do so is to talk to more people. Learn their perspective and try to empathize with them. As the cliché goes, you walk a mile in their shoes. This is why many people who travel to other locations and cultures tend to have an easier time with abstract thinking, as they’ve exposed themselves to many new ideas, perspectives, and ways of life.” says Thomas.

What this means is that abstract thinking - similar to critical thinking - is a skill. It needs to be practiced and, in math terms, this means the more you do it, the less arduous it is. But, this makes it sound simpler than it is because few people want to sit down to practice math. Let’s discuss why that is.
**Our brains are wired to be preferential towards emotions and convenience. **

When I say emotions, I don’t necessarily mean basic emotions like happy, angry, or sad - though those can play a role - instead I mean psychologically our brains are wired to do things that make us feel good and avoid things that don’t. Unsurprisingly, for most people who struggle with math it comes down to how it makes them feel. Those feelings can vary wildly from boredom, to feelings of frustration or a lack of control, to a feeling of self-surrendered exhaustion; and to feel those emotions would make anyone, understandably, not want to engage in a subject or task.

K. Abdul Kafoor & Abidha Kurukkan’s paper Why High School Students Feel Mathematics Is Difficult? An Exploration of Effective Beliefs interviewed a sample size of 51 high school students and their results add credence to this psychological phenomenon, to paraphrase: math was the most liked subject for 6% of students while it was the most hated subject for 88%.

Liking it played a role in student interest, boredom, self-efficacy beliefs (which is a fancy term for your perceived confidence in a subject and how that plays on your ability to perform it, like how Ron in Harry Potter and the Half-Blood Prince played quidditch really well when he believed Harry had given him the luck potion - his self-efficacy belief had skyrocketed).

Students who viewed or felt that math was difficult also tended to dislike it more than those who said that it was easy. And to quote the survey, “Students’ likes towards mathematics in turn [was] significantly dependent on their self-efficacy in mathematics. That is, students who [liked] mathematics [tended] to have positive self-efficacy [at 89%] and those who [disliked it tended] to have negative self-efficacy [at 53%]” (Gafoor, 2015).

As it turns out, not only do people not want to do things they dislike doing, but it also makes something that is disliked more grueling to do. And unsurprisingly, hard things are inconvenient, which our brains also dislike engaging in.
**So what about convenience? **

Basically, our brains love convenience and they love taking shortcuts. The reason our brains love that kind of thing is because it reduces processing power and saves energy. It’s the same reason Steve Jobs was quoted to always wear the same outfit everyday; and similar to actively designing an outfit in the morning, math requires energy to do - everything requires energy to do - and sometimes we want to utilize that energy for something we deem as more valuable.

So for example, let’s say I asked you what 1 + 2 + 3 + 4… + 99 + 100 is equal to, without looking it up, what is the answer?

Regardless of how you got the answer, I’m guessing you didn’t grab a piece of paper and painstakingly add up each individual number. In all likelihood, you found some sort of shortcut to make the problem far more convenient, or to save maximum energy, you didn’t find the answer - it’s unimportant to your life and not worth the energy.

Math can be wildly inconvenient, if a shortcut isn’t available. And if we have to actively put energy towards solving the problem then that can be legitimately exhausting, especially if that’s not what you’re used to doing nor what you want to do.

But that can be compounded too if, for instance, you’re doing complex math and weren’t taught the proper foundation to do it. For example, you can’t do trigonometry if you weren’t properly taught algebra.
**So, to my third point, did your teachers fail you? **

Were you not properly taught to do math and that’s why it’s difficult? Well, actually, it could be. “[Math is] a language that too many people never learn, often because the education process misses the number of ways that a given person can arrive at a given solution. Part of the challenge is to identify the gaps in knowledge, to clarify that the challenge is not that a student simply doesn’t understand algebra or trigonometry or whatever. There may be a particular basic concept that stands in the way of going forward in math” (Crow, 2016).

To lend a first hand account to this: I have a rough time with deducing percentages because I wasn’t properly taught fractions. The question, “If two people flip a coin, what is the percentage of time that both coins will end up heads?” confused me initially. My solution was to add ½ (50%) since there’s only two sides to a coin and you can only get one outcome, with another ½ (50%), and that doesn’t work because obviously you can’t get both coins to be heads 100% of the time.

After asking a friend, they said the answer was ¼ of the time, or 25%, and they taught me this by not using math. Instead, they said, the outcomes are “Head & Head, Head & Tail, Tail & Head, and Tail & Tail, meaning of all 4 potential outcomes, the one we want can only happen a single time.”

Grace Fleming, M. Ed., says it best, in her article, Why Math Is More Difficult For Some Students, “Math know-how is cumulative, which means it works much like a stack of building blocks. You have to gain understanding in one area before you can effectively go on to ‘build upon’ another area. Our first mathematical building blocks are established in primary school when we learn rules for addition and multiplication, and those first concepts comprise our foundation. The next building blocks come in middle school when students first learn about formulas and operations. This information has to sink in and become ‘firm’ before students can move on to enlarge this framework of knowledge. The big problem starts to appear sometime between middle school and high school because students very often move on to a new grade or new subject before they’re really ready… So students move to the next level with a really shaky foundation. The outcome of any shaky foundation is that there will be a serious limitation when it comes to building and real potential for complete failure at some point.”

It’s not necessarily the fault of your teachers - they tried their best - but the curriculum is often not conducive to a proper understanding of certain subjects and you could be missing a key, foundational understanding that was meant to be taught to you somewhere along the way and either wasn’t taught at all or was taught, but not well enough.

Which leads me back to the question: why is math hard? It’s abstract and that can be perplexing to comprehend if one is unpracticed in that style of thinking. Also, people can tie negative emotions to it, making them not want to do it, on top of it requiring energy that could be put towards something more important. Lastly, it could also be difficult due to a lack of key mathematical foundations that weren’t properly touched on in school. And now, we know why math is hard.

**Singh, M. (2021). Is Math An Absurd Subject? **
https://numberdyslexia.com/is-math-an-abstract-subject/
**Thomas, J. (2022). The Difference Between Concrete Vs. Abstract Thinking**
https://www.betterhelp.com/advice/self-esteem/the-difference-between-concrete-vs-abstract-thinking/
**Gafoor, K. A. & Kurukkan, A. (2015). Why High School Students Feel Mathematics Is Difficult? An Exploration of Effective Beliefs**
https://files.eric.ed.gov/fulltext/ED560266.pdf
**Crow, M. (2016). Why Is It So Hard to Learn Math?**
https://www.linkedin.com/pulse/why-so-hard-learn-math-michael-crow/?trk=mp-reader-card
**Fleming, G. (2019). Why Math Is More Difficult for Some Students**
https://www.thoughtco.com/why-math-seems-more-difficult-for-some-students-1857216#:~:text=Math%20seems%20difficult%20because%20it,to%20collapse%20at%20some%20point.