Math Working

[Jacquelyne Betance]

Ireally thought about that for several days. I agree about the importance of understanding what students are thinking and for them to be able to communicate what they are thinking through their work.

Now that most of us are face to face, I have had an opportunity to see more of their work. After learning virtually for several weeks, many students got in the habit of showing little to no evidence of the work they did to reach their conclusions. In general, they were out of the practice of writing anything down.

To start with I needed to determine what I consider to be good math work. What do I like to see when grading, and what helps me when I am personally working math problems? I decided that characteristics of good work would be:

My health started to deteriorate. I gained 20 pounds in three months. I developed insomnia. I went back on antidepressants. Taking a step forward professionally meant several steps backward in every other part of my life.

There was a moment, two months into my beleaguered balancing act, when I broke. It was in the bathroom stall of a concert venue in Worcester, Massachusetts. My then-boyfriend, two pals, and I had packed into a car for the two-hour drive up, to see the final performance of a hardcore punk band, Bane.

Maybe it was the stench of caked urine. Or the guttural wails from the stage, shrieking through my body. Or the weeks of working without pause. My heart thumped, my limbs wobbled, my head whirled as my stomach went hot. I spent maybe 15 minutes in the stall, but my experience of time suffered one of the defining talents of panic attacks. Time became a vacuum.

I'm a newcomer to Latex and have been relying on lots of examples to learn how to format a math proof. I have been really stuck on this particular part of the proof and was unable to find resources or a similar example to fix:

I just installed RankMath on a new site. The site was already using a plugin called Redirection to manage my redirects. When I installed RankMath, the Setup Wizard told me that it would import my redirects from Redirection, as well as deactivate Redirection, because the plugin was incompatible with RankMath.

However, while Redirection was deactivated, RankMath did not import ANY of my preexisting redirects. The RankMath redirections page is completely blank. So I have to set up all my redirects again from scratch.

My main redirect was a regex that converted old WordPress URLs that used a date structure in the URL to just the post type in the URL. I recreated the same source and target URL regex code in a new redirect in RankMath, but it is not working. This exact same code worked just fine as a redirect in the Redirection plugin. Can you tell me what needs to be changed for RankMath to use it correctly?

I apologize for the inconvenience you are facing with the redirection regex in Rank Math. It seems like the regex you used from the Redirection plugin needs a slight adjustment to work correctly in Rank Math.

In the Redirection plugin, I could use /$1 as the destination URL, which matches the format of the source URL (the exact string to replace). However, RankMath forces the domain into the field, and I am wondering if that might be why the redirect is failing.

The main thing I feel is that I'm not organizing my mind and my derivations as clear as I could, because I don't have the best "math habits". I feel like if I could develop better math habits, I could significantly improve both my time efficiency and the quality of my thinking.

To show what I mean, I'll compare it with the skill of writing: I used to write in a very unstructured way: I simply started writing with some vague idea of what I wanted to write. Then after having written a paragraph, I would generally be somewhat confused. After 2 paragraphs I'd be more confused. Eventually I didn't have a clear idea of what to write because my mind was so cluttered, as if all my neural pathways were firing un-synchronously, creating a senseless mess. I have now solved this by developing better habits: I started making bullet point lists of my papers that contained the central argument, before I wrote the actual paragraphs. I then wrote one paragraph at a time, focusing only on what that particular one had to convey. Also, I developed a more structured way of structuring paragraphs: rather than just "writing it", I thought about the first sentence separately, and then its relation to the second, and so on... After developing these better habits, I felt like my brain had a much more "lean" and "uncluttered" process it was following, as if my neural pathways fired synchronously, in harmony.

I feel like right now with maths, I am in a similar stage that I used to be with writing. I understand math concepts, and I know how to do many of the methods, and I'm progressing. But whenever I'm working on a math problem, I feel like I'm getting confused, not just because the problem is new and difficult, but because my mind is cluttering and confusing itself, as if I don't have a "process" that is optimized for figuring out new math.

One way this shows, though I don't know if its a cause or a symptom, is that my derivations look like a plate of spaghetti. Yet if I try to write things more structuredly, I'm held back even more, because it puts me into a very "fearful" and paralyzed state of mind (fearful to write something wrong).

So I'm looking for habits that I can develop that will, just like I did with my writing process, turn my "cluttered" mind, into a "harmonic" one. That doesn't mean math will suddenly be easy, but at least the difficulty will be due to the complexity of the math, rather than due to me working against myself.

So I'm interested if any of you have experienced this same thing, and whether there have been specific habits or other things that have helped you overcome this.

To give an example of something that recently has actually helped me somewhat: Whenever I now derive an intermediate result, I write big boxes around it, with a big dense filled circle in the corner, in order to signify that it is an important result. This somewhat declutters my mind, because I no longer have to wade through all the intermediate steps, looking for the important stuff.

ps. I hope this question is not too general or subjective. I know that subjective questions are not the purpose of math.stackexchange, but I thought: there certainly are some objective principles behind what kind of habits work and don't work. And I wouldn't be surprised if I'm not the only one who could benefit.

I think this is a great question and you've already made an important step in addressing the problem - realizing that you are not satisfied with your math working process and searching for ways to improve it. Here are some ideas and suggestions which I found helpful:

Play with simplified models. This is something I really learned in graduate school and I wish I would have been told explicitly much earlier. If you are facing a problem that you have no idea how to approach and you feel paralyzed, try to work on a simplified (even trivial) model. For example, let's say you need to prove some statement about a linear map $T$ on some vector space $V$ and you have no idea what to do. Can you solve the problem if you assume in addition that $V$ is one-dimensional? Even better, if $V$ is zero-dimensional? Can you do it if $T$ is diagonalizable? If you are asked to prove something about a continuous function, can you do it if the function is a particularly simple one? Say a constant one? Or a linear one? Or a polynomial? Or maybe you can do it if you assume in addition it is differentiable?

Applying this idea has two advantages. First, more often than not you'll actually manage to solve the simplified problem (and if not, try to simplify even more!). This will increase your self-confidence and help you feel better so that you won't give up early on the harder problem. In addition, the solution of the simplified problem will often give you some hints on how to tackle the general one. You might be able to perform an induction argument, or identify which properties you needed to use and then realize those properties actually apply in a more general context, etc.

EDIT: I misunderstood the OP at first, and the first half of my answer gives advice on how to approach proving an unknown problem. I then tie this into the organizational question the OP is really asking below the line.

Every time you see a theorem, first seriously commit yourself to finding a counter example. Find almost-counterexamples that show why every assumption in the problem is necessary. Then for each of those almost-counterexamples find an example that is extremely similar, except satisfies the assumption the counterexample was missing. Now you're ready to prove the theorem or read its proof, and in all likelihood you're already close to the proof.

One day in his office, I happened to mention Bezout's theorem which basically says that two curves of degree $m$ and $n$ respectively intersect in $nm$ points. He says he never heard of it and seems galvanized by it. He jumps up and heads to the blackboard, saying "Let's see if I can disprove that" Disprove it?! "Wait a minute!" I say, "that theorem is nearly two centuries old! You can't disprove anything... really..." As he begins to working on some counterexamples at the blackboard I see my well-meant words are simply static.

His first tries were easy to demolish, but he was a fast learner, and ideas soon surfaced about the complex line at infinity and how to count multiple points of intersection. After a while it got harder for me to justify the theorem, and when he asked "What about two concentric circles?" I had no answer. He argued his way through and eventually found all four points. Finally he was satisfied, and the piece of chalk was given a rest. He backed away from the blackboard and said "Well, well - that is quite a theorem, isn't it?"

3a8082e126