# Mechanical Versus Conceptual Difficulty

## On finding gaps in your STEM education and enhancing your lifelong learning

It’s a tragedy whenever someone gets that degree or lands that job, and puts their days of earnest learning behind them. The truth is, we come away from our formal educations with holes in our abilities. Often, these gaps can be filled in a way that enhances how we perceive the world around us, with only a small investment of time and attention.

I believe that one class of learning gaps comes from the mismatch between the mechanical difficulty and the conceptual difficulty of the things we learn in school. Often, our studies cause us to misallocate our efforts into overcoming one form of difficulty over the other.

In this article, I will briefly define what I mean by mechanical and conceptual difficulty. After that, I will talk about where you might encounter different combinations of difficulty and what strategies might be useful in each scenario.

Note that this is written primarily with the sciences and mathematics in mind.

## Mechanical and Conceptual Difficulty: the “How” and the “Why”

A problem is mechanically difficult if it takes many steps to solve, requires lots of intermediate information to be stored, or has many possible actions to be tried.

At the very easy end of the spectrum we have things like arithmetic. People generally memorize single digit multiplication, so it takes 1 step to solve, requires no intermediate information, and is a single act of recall. Multiple digit multiplication is slightly more difficult, to the point where it’s helpful to follow a simple algorithm and write down a few intermediate digits.

Broadly speaking, the mechanical difficulty of a problem is tied to *how* the problem is solved.

On the other hand, conceptual difficulty is related to the ease with which one can answer the following questions about a problem:

- What is the purpose of doing this?
- In what situations would I expect to encounter a problem like this?
- How is this similar/dissimilar to other things I’m familiar with?
- Can I explain what I’m doing in words?
- Are there different but equivalent ways of looking at this?

A topic is conceptually deep if there are many ways of thinking about it, or if the motivation for it is built upon many other concepts. Part of understanding something conceptually is understanding *why *a problem is worth thinking about and learning in the first place. Mastery of this aspect will allow you to make broad analogies and bring problem solving strategies across into diverse domains.

## Examples and Strategies

The above is my personal difficulty ranking for a few topics I encountered around the time of high school and university. Keep in mind that a good amount of subjective judgment went into this. Also, even if everything seems easy to you after you learn it, it does you no good to feed your hindsight bias, so I’ve ranked based on difficulty at the time of learning. Finally, I understand that many topics can be understood very deeply even if their barrier of entry is small, so please don’t get offended if I ranked something lowly.

In any case, *how* you rank various concepts is less important than the fact that you *do* rank them, because as we will see, you can approach topics in each quadrant with a tailored mindset and strategy.

**Mechanically Easy, Conceptually Easy**

I’ve already talked about arithmetic as falling into this category. Simply put, concepts in this category can be learned in a matter of days.

The only real mindset you need regarding this category is one of openness. Don’t assume that because things are easy to learn, you must have been taught them already. Recently, I encountered “hat puzzles” and was amazed that interactions like this could take place:

*Person A: I don’t know the color of my hat.*

*Person B: I don’t know the color of my hat.*

*Person A: I know the color of my hat.*

Even though it all comes down to understanding the single proposition that *the admission of a lack of information is itself information*, that idea is a stimulating one. If I had not been open to the fact that I had not yet thought about a simple idea, or if I had dismissed it as trivial after the fact, the pleasure of learning would have been diminished.

While simple concepts can be grasped in days, or even minutes, don’t assume you’ve encountered them all.

**Mechanically Easy, Conceptually Difficult**

This is the quadrant where, in my experience, those “gaps” left by formal education are typically residing. Schools often emphasize mechanical mastery over conceptual, so they might move on from a topic before students fully grasp the implications of what they’ve learnt.

In fact, the subject of calculus as it’s often taught in schools is a way of bypassing subtle geometrical thinking by turning it into symbol shunting rules.

When dealing with concepts from this category, be on the lookout for types of problems you *think* you’ve mastered because you know how to solve them. Once you’ve identified topics where your conceptual grounding is fragile, try to express your understanding in as many forms as possible. These may include:

- Notation
- Plain words
- Diagrams
- Code

Learning with the intention to explain to others (the Feynman technique) can be especially helpful here.

**Mechanically Difficult, Conceptually Easy**

I call this the “quick wins” quadrant. Are there problems you can think of that you really struggled with in school? Chances are that for at least some of them, the motivation behind them and what they’re useful for is easier to grasp than the actual process of solving them.

For me, Taylor series were a pain to deal with in high school. But consider this informal explanation:

*If you know where you are, how fast you’re moving, how fast you’re accelerating, how you’re changing your acceleration, and so on, then you can estimate where you’ll be in a few moments. The quality of the estimation will depend on the highest order rate of change you include, and the time from now you want to know your position.*

To me, that explanation satisfactorily explains the goal of Taylor series in the context of position as a function of time. Not only that, the general idea of forming an estimate by summing higher and higher order adjustments is broadly useful.

The best thing about this category is that mechanically difficult problems can often be solved easily using software, and that a lot of these tools have already been developed by other people. Doing things by hand is rarely necessary. So go ahead and revisit topics that you struggled with in school, and see if they have a simple motivation that can be useful in your broader problem solving efforts.

**Mechanically Difficult, Conceptually Difficult**

Concepts in this quadrant are the hardest to learn, but also the most rewarding.

One of my favorite essays is Teach Yourself Programming in Ten Years by Peter Norvig. What attitude do I think you should approach both mechanically and conceptually difficult topics with? Gratitude. It’s an amazing thing that there exist ideas that can capture the imagination and be digested over many years, even as many as ten or more. They are a testament to the richness and complexity of our universe.

So when you find something in this category, don’t give up and appreciate the journey. Reframe your understanding in as many ways as you can, just as with the mechanically easy and conceptually difficult problems, and don’t stress being able to solve things by hand so much, as with the mechanically difficult and conceptually easy.

## Parting Thoughts

In this article, I’ve shared a framework I use to find learning opportunities, and determine what attitude I should approach them with. The framework is based on the idea that mechanical difficulty and conceptual difficulty don’t always go hand in hand.

To summarize:

**Mechanically Easy, Conceptually Easy:**there are more of these than you think. Be open to the idea you haven’t encountered them all.**Mechanically Easy, Conceptually Difficult:**this is where you’ll find gaps in your education. Try to explain things in many different ways.**Mechanically Difficult, Conceptually Easy:**you’ll find quick wins here, especially because mechanical difficulty can be circumvented with software.**Mechanically Difficult, Conceptually Difficult:**be thankful that these exist. Don’t try to take too many shortcuts.

I hope you find this way of thinking useful, and that it leads you to perhaps view things you’ve given up on or overlooked in a new light.