How can I improve my comprehension of mathematical notation and avoid getting lost in the details, when there's no obvious visualization to rely on?

Question

For context, I'm an engineering student nearing the end of my degree and I've been tutoring algebra, calculus, and statistics for years, so I'm not a newbie to math. It's only really been an issue in the past year or so as I've taken more theory intensive classes involving lots of long derivations full of algebraic manipulation and calculus tricks -- I found myself skipping over them entirely because it feels too overwhelming trying to parse through all the notation, even though it's usually notation I'm plenty familiar with and have no issue with in smaller doses. Let's take this as an simpler example that I was just a little while ago having the same issue with:

when I see the differential equations, my instinct is just to skip over them, even though doing that means the textual portion of the article won't make much sense.

I can make myself slow down and mentally parse the syntax, but doing so can be mentally exhausting if there's a whole lot lines similar to lines 2 and 3 in the above example. I think at least part of it is that mathematical notation is simply a lot more information dense than English, and so trying to mentally "pronounce" all the symbols or even a verbal description of a larger "word" (e.g. "d squared x squared of p naught u" vs "second derivative with respect to x of p naught u") results in a lot more mental "speaking" than the amount of space taken up on the page would imply. That, for lack of a better word, "misleading" visual makes me associate it with an inordinate amount of mental effort.

I had asked a friend with more mathematical maturity than me for advice and he said that he sees the notation as more analogous to diagrams than letters and words, so he doesn't usually try to mentally "pronounce" it. I tried thinking of it that way, which seemed to help at first, mostly because it helped me focus more on larger expressions as a whole, instead of trying to parse each individual piece, but I have a hard time really comprehending the information that way unless I mentally describe the "diagrams" to myself (something like, "ok, a second derivative, a first derivative, u, coefficients of p_i equal to squigly L u of x") which doesn't save that much mental effort. But more than that even, I often get so focused on trying mentally parse/describe the notation that I loose sight of the bigger picture, of what it is that's being derived or the like.

Oh, I should probably also mention that I don't typically have trouble with writing long derivations and calculations -- I'm actually pretty competent with LaTex and I regularly do my assignments using it (those I use an extension that shows me preview of the rendered version in real time, so I can more easily see what I'm doing). I'll also often write out sentences describing/explaining each step in words. I think the reason I struggle with reading such things but not writing them is that, when I'm writing out my own work, I often already have the big picture strategy mapped out in my head, plus, it's emotionally satisfying to translate the general outline in my head to a neat and organized step-by-step solution on the screen. Also, many engineering textbook derivations are almost entirely just algebraic manipulations and the like, with very little explanatory text interspersed, whereas I almost never write out my own work like that -- I either write the purpose/reasoning for each non-obvious (to me) step in a separate sentence/phrase, and/or if I'm doing the work in code (I frequently make use of Python's NumPy and SymPy libraries to speed up calculations and derivations), I name the variables and/or functions very descriptively, which serves essentially the same purpose.

Finally, I did see this question, which is similar to mine, but the two helpful answers (the last one basically just says "practice makes perfect") focus on visualizing the concepts the symbols represent. And, while I completely agree doing that can be incredibly useful in aiding comprehension, and I actually do it instinctively when I have an idea of how to do so. But when it comes to something like differential equations, which is what most of the derivations in my classes have focused on recently (mainly the Schrodinger equation and Maxwell's equations), there typically isn't any obvious visualization for all the steps. Even though I often have a mental picture of the quantities represented by each variable (e.g. position, EM field, electron energy level), that doesn't really help with comprehending all the different operators being applied or all the different manipulations being done to the equations, not because I don't understand the individual operators or the algebra or even because I don't have mental pictures of what they mean (sometimes I do, sometimes not), but because I don't have a picture/visualization of how everything fits together in long and complicated derivations and trying to come up with one for every single derivation would take an inordinate amount of time.

How can I improve my comprehension skills in this regard, so that it's not such a discouraging mental workload to read even small derivations or other mathematical content and so that I don't lose sight of the bigger picture, missing the forest for the trees? Should I try to come up with visualizations for everything? Or is there a better way?

Answer 1

This question seems hard to answer in general, but I suppose you are looking for helpful suggestions rather than a generally applicable solution. So here are some suggestions that sometimes work for me when a mathematical text is overwhelming in terms of formulas etc.:

1. Read the text several times. The first time you can skip all the details but try to understand the overall message. The second and third read will become easier and reveal more understanding as you delve deeper into the corresponding formulas, which become less and less repulsive. Repeat until you are satisfied with your understanding (which is not necessarily the same as getting all the details).

2. Break things down into smaller pieces. As with any overwhelming task in life, it helps to divide it into smaller tasks. Take breaks between those tasks and don't be too hard on yourself. These things take time. :-)

3. Often, I find it really helpful to write down the formulas and derivations myself on paper, rather than just reading them. This slows down the process of reading these "dense" formulas and helps to wrap my head around the concepts at my own pace (and also to remember them). I guess this depends somewhat on your learning style.

4. Sometimes, the formulas can be simplified, so this might help as well. If you find that the book/paper you are reading has many formulas/explanations that are unnecessarily complicated and can be simplified, you might want to switch to a different book/paper/website. Having several sources can be helpful, but can also be overwhelming, so it is important to find a good balance that suits you.

5. How well do you understand certain parts of the formulas? You study Maxwell's equations, so you should have a good intuition for divergence, curl, gradient etc., which are the underlying basic concepts. Sometimes one has to go back and first study those again. The good news is that there are great visualizations for these - you don't have to do everything yourself. Have a look at this video. With this in mind, it might be easier to understand more complicated formulas and they are less off-putting.

6. Try to visualize the concepts whenever possible. As you wrote, this can be time-consuming, so don't waste too much time on this and don't necessarily try to visualize everything. Also, maybe someone has already put in the effort and you can find good visualizations online.

7. When it comes to mathematical theorems, understanding simple and less simple examples and counterexamples is extremely valuable. This might be less important for you as an engineer, which is why I put it as my last point - for mathematicians it should be one of the first. This can sometimes be time-consuming, but it really helps to get a good intuition for the theorem. When I don't get the statement of a theorem, I try to find a counterexample. Naturally, I fail, but it gives me an intuition why the theorem might work even without reading the proof. It also might give me an idea on how the proof might go, so it could be easier to read the actual proof afterwards - the formulas will be less overwhelming. My favorite example for this is the statement "Every continuous function on a compact set attains its maximum value". Trying to come up with a counterexample by actually drawing functions on, say, $[0,1]$ really helps your understanding, as well as finding counterexamples when you drop certain assumptions (continuity/compactness). Also, studying simple (but non-trivial) examples is often helpful.

Hope this helps.

Answer 2

It comes with time, and real effort.

You have to persist and reread everything as many times as needed. Of course, you can try to come up with visualizations (this is entirely a personal matter). But that is difficult (and again a personal matter) for highly abstract concepts.

For any trained mathematician these formulas and any long(er) derivations become part of the language, and in fact even very abstract concepts often become quite natural. The preference, probably unsurprisingly becomes the formulas themselves. How could you be rigorous otherwise without having to write extremely long sentences?

A suggestion that I would make is that you have a look at some book that provides context, at a sufficiently high level. One of my favorites is Lars Garding, Encounter with Mathematics. There are many more. This would help visualize a much bigger picture.