“I am confused…”

From learning math (more or less) for the past sixteen years, I came to an interesting realization: I learned so much from being confused.

In this post, I will discuss different kinds of confusion based on my personal experience.

Different Dimensions of Confusion

I consider three dimensions of my confusion: correct/incorrect understanding, confused/not confused, should/should not be confused. I will explain more about what I meant by each dimension more carefully in the following sections with specific examples.

Correct Understanding; Not Confused; Should Not Be Confused

The concepts fall into this category are mostly mechanical concepts, for example, how to count/add/multiply numbers. Taking adding numbers as an example: my understanding of the concept is correct (e.g. I can correctly add numbers); I was not confused (e.g. I didn’t have trouble with doing the computation); and I don’t think I should be confused (e.g. although I might change my mind in the future, I don’t believe it’s super interesting to ask “why do we add numbers in this way”?).

There’s a surprisingly low number of concepts fall into this category, and most of them are those I learned from primary school.

Correct Understanding; Not Confused; Should Be Confused

The existence of this category brings me so much joy during learning! This category broadly consists of concepts such that my understanding wasn’t wrong per se, and I didn’t have any question about the concepts, but after learning more, I regret not being confused when I first learned it.

Let me give an example. Matrix multiplication falls into this category. When I first learned it, I was able to correctly do the computations (i.e. correct understanding), and I didn’t have any question about the concept. Later in linear algebra, I learned that each matrix is a linear transformation, and that matrix multiplications correspond to compositions of transformations. It wasn’t until then that I realize, how did I just take the definition of matrix multiplication as granted in the first place?! In hindsight, the multiplication rule should’ve seemed unnatural to me: why can we multiply a 4-by-3 matrix with a 3-by-2 matrix, but not with another 4-by-3 matrix, despite them having the same size?

Most of the concepts in this category are those I learned from middle school and high school. Concepts in this category are those that seem “trivial” as first (and that’s why I wasn’t confused), but have deeper meaning. Whenever I discovered a concept falling into this category, I feel like I’m learning — after all, it’s relatively easy to teach someone how to multiply matrices, but it’s harder to explain why the rule is defined this way, as an explanation involved more advanced concepts such as vector spaces and linear transformations. This category is also closely related to the idea of asking good questions; as a result, I believe it’s an important skill to have less items in this category, and more items in “correct, confused, should be confused”.

Correct Understanding; Confused; Should Not Be Confused

I don’t believe any item should belong to this category :). In general, I believe whenever I am confused, realizing the confusion is extremely important. Especially if I am confused despite my seemingly-correct understanding, the confusion can turn out to be valuable, because it hints that there’s something deeper going on.

Correct Understanding; Confused; Should Be Confused

I am very proud of this category. It might seem odd to say I am confused despite having a correct understanding. It turns out that this actually happens pretty frequently.

Although I mentioned that I took the definition of matrix multiplication as granted, I did get confused by the definition of the determinant of a matrix — it wasn’t like I didn’t know how to compute determinants, I just didn’t get why we define determinant in such way. I was later taught to view determinant as a volume scaling factor, but I have to say that I sill remain confused about determinants, and hopefully my confusion will clear some day.

In the above example, my confusion was well-defined: “why is determinant defined like this?” seems to be an answerable question. For a lot the cases, I couldn’t even define my confusion! I just knew I was confused. For example, when I first learned modular arithmetics, I can do the problems correctly, but I just felt like there was something missing. If I were to ask my instructors, my question would’ve been “I just don’t think I really understand it”, which is a rather vague question that’s not really answerable. I have seen an accurate analogy: it was like being in a dark room; I can probably get a rough sense of what the room is like by touching around, but I don’t really know what the room looks like. A year later, I took abstract algebra, and it was a fascinating feeling — it feels like I finally found the switch in the dark room, and turned on all the lights.

This category most consists of concepts I learned in college, plus some from high school. I am always glad whenever I found such concepts: if I can phrase my confusion, that means I can ask questions and hopefully get a satisfying answer; if I couldn’t phrase my confusion, that probably means my brain senses there’s something deeper and seeks a more abstract framework to organize my knowledge. For most of the cases, I am able to clear the confusion after relearning the concepts in a more abstract way, but I have to say that I’m still confused about a lot of the concepts in this category, and I’m looking forward to the excitements after I finally figure them out!

Incorrect, not confused, should not be confused

In my opinion, it is very dangerous to have an incorrect understanding and not being confused. As a sanity check, I have never had any experience of not getting a satisfying grade in a course (a naive way of justifying the “correctness”) that I feel like I really understood everything. On the other hand, whenever I didn’t do well on any test, I knew I wasn’t going to do well.

Incorrect, not confused, should be confused

Same comment as above.

Incorrect, confused, should not be confused

As I discussed in the “correct, confused, should not be confused” category, there’s no item in this category.

Incorrect, confused, should be confused

Items in this category are relevant in terms of performance in classes. Incorrectness, normally reflected by getting a problem wrong on homework or exams, is a good way to identify confusions. It is particularly helpful when I didn’t know how to phrase my confusion: if I got lucky, correcting an incorrect answer is exactly the solution to my confusion. I regard items in this category as confusions that are available for quick fixes. Examples in this category include algorithms that I didn’t learn properly at first.

Discussion

I want to take some space to discuss my view on the difference between being correct and really understanding something. Before college, I used to be obsessed with correctness: am I doing the computations correct? am I getting a good score on the tests? And then I started college and didn’t even know what it means to be correct anymore! Does correctness mean that I understand the statement of some theorems? But I don’t think I can say I’m “correct” until I understand the proofs. What if I understand the proof? But I think I have to be able to demonstrate my understanding, at least by solving homework and exam problems “correctly”. But even if I was able to answer those problems, now that I graduated, so what?

Four years of college taught me the valuable lesson that it’s much more important to turn my focus to gaining a deep understanding of concepts. It takes time, and I frequently feel frustrated, but it also feels great whenever I clear a confusion. It really feels like turning on the light in a dark room that you’ve been longing to look around.

Still, I included correctness as a dimension in the above analysis of confusion. As I probably finished my last math class, and that with self-learning, it’s likely items would pop up in the category of “incorrect, not confused”. I need to make sure this category doesn’t grow bigger.

Being confused is okay. It is an important component of learning. I wish myself that during my future studies I can still occasionally be confused, and constantly have a curious mind.

A Scary-Looking Quotient Ring …

I was preparing for my algebra final and I ran into this quotient ring constructed in $\mathbb{Z}[\sqrt{-5}]$:

$(1 + \sqrt{-5}, 3) / (1 + \sqrt{-5})$

What even is this?!

Why Is Many-One Reducibility Not A Total Order?

We use the notation $A \leq_m B$ if there’s a many-one reduction from set $A$ to set $B$. When I first learned about many-one reduction, the notation $\leq_m$ made me think $\leq_m$ imposes a total order among subsets of $\mathbb{N}$, i.e. $\leq_m$ on $\mathcal{P}(\mathbb{N})$ behaves like $\leq$ on $\mathbb{N}$. Unfortunately, $\leq_m$ only imposes a partial order on $\mathcal{P}(\mathbb{N})$. In this post, we will discuss an example of two subsets of natural numbers that are not comparable using $\leq_m$.

Integral Domain

We covered Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains before Thanksgiving break. I was slightly stressed during the break because I didn’t grasp the concepts very well. Luckily, when I get stressed, I write notes, and then I’ll be happy again 😉 So here it is:

Let’s Partition A Group!

I think of a group as a bag of items with a rule of smashing two items into one. Sometimes, we don’t want mix everything together in one bag; we might want to divide the items into several smaller bags. Let’s talk about two ways of partitioning a group, i.e. dividing the items in the bag.

Shuffle, Shuffle, Shuffle …

Groups are like verbs. We can gain more information about a group by looking at how it interacts with other objects.

What Does Order Tell You?

When I first learned algebra, I thought there’s no reason to believe we can gain structural information of a group solely by looking at the number of elements of the group. Unfortunately (or maybe fortunately :p), I was wrong.

Random Coin

Let’s play a game. I toss a coin twice and ask you, what is the probability of landing two heads? I claim that my coin is a fair coin. Considering I might be lying, what probability should you guess?

The Isomorphism Theorems

We learned about quotient groups and four Isomorphism Theorems in my algebra class. My professor refers the four Isomorphism Theorems as “four siblings of Isomorphism Theorems” (of course, that made me laugh; it always makes me happy when people describe math in lively terms). I’ll write about my understanding of them in this post.

An Idealized Programming Language

Imagine this: you write a program, compile it and you start to run the program. It has been running for one minute (“this is not a incredibly fast program”, you think;) five minutes (“oh, it’s kind of slow”), one hour (“okay it’s very slow”), one hour (you start to become impatient) and after one day, it’s still running. You start to wonder,

“would it ever stop running?”