Goals and approximations

It’s been a good week for reading. I’m about two-thirds of the way through Tombstone, picking up the pace in The Development of Mathematics, and I finished one biography of a mathematician (the charming and generous George Pólya) and started on another (the eccentric Paul Erdős). In Mathematician’s Delight I got right up to some problems I really wanted to take a crack at solving before every other deadline in the universe seemed to come crashing down upon me. My weekend was an attempt to meet as many of these deadlines as I could, and I hope that I did my best.

Now, here we are on the cusp of July — June always seems to pass in the blink of an eye — and with the new month will come a new set of priorities. Plan for a trip to Ohio and West Virginia and back again. Create a writing weekend. Donate a few carloads of items that I don’t need. Start working on my cardio. Reinvent my identity. Practice my Hebrew. Get my father’s photos scanned. Lay out another month of my bullet journal. Keep reading. Go to a local concert for the first time in about 18 months.

What did I actually do this weekend? Attend online services, exchange texts with my mother, drive my son to work, shop for groceries, attend my final Torah cantillation class, help to prep beef stew, do laundry, do dishes, pick up my son from work, fill the gas tank, shop for books, pick up some free custard, watch qualifying for the Styrian Grand Prix, lay out several pages of the congregational newsletter, attend a retirement picnic, transfer a loaner car from one loanee to another, drive my son back to his father’s house, get pulled over by a deputy sheriff for non-display of plates, drive home, locate and install my front license plate, do more dishes, watch the Styrian Grand Prix. For some reason it does not seem like enough. I intended to relax on my birthday, but didn’t relax enough. I intended to get things done the next day, but didn’t accomplish enough.

What is enough, really? Do we ever catch up? Lately I’ve been chasing the goal but it’s tough to tell whether or not I’m making progress. Sometimes the proof is external and sometimes it’s internal. Externally, perhaps I did all of the things I could do. Internally it “hits different,” as my kids say.

This summer will be one of transition, for many people in my immediate circle. Transitions aren’t easy and we’ll need to find a way to show each other compassion and grace as we move through our changes. What can I say? Show your love to the people you love. Maybe that’s enough.

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Knitwise, I haven’t been doing any Actual Knitting™ in the last week but I did come across a link to a free download for an Interweave e-book that contained two patterns I want to make. One is for a triangular shawl made with a bit of openwork but NO PURLING, and the other is for a scarf/stole with a simple lace panel, made with bulky yarn of which I seem to have a sufficient amount. So if I find the time to knit, I’ll be ready to cast on.

Painless

Several years ago, a friend of mine described mathematics as an anodyne. I had to look it up, and anodyne means something that relieves pain, particularly the pain of life. (Death, in the 1500s, was also described as an anodyne, but let’s not go to extremes.) I realize that to most of my friends mathematics, particularly the kinds that include letters, feels like the reason for the pain and not the cure of it. But bear with me for a moment.

I’m fortunate enough to know people who absolutely love to fix cars, including my car. I’ve even more fortunate to know people who love knowing the ins and outs of the United States Tax Code so well that they can find me a refund every year that allows me to pay the guy who loves to fix my car. It’s a win-win-win scenario. If, on the other hand, I had to be the one to do the wrenching and the tax-code referencing, I might end up tearing out all my hair. At tax time (and tuneup time) I’m particularly glad that there are technical specialists who get to make a living at what they love to do.

So even if you dread math, please give a pass to the math teachers and mathematicians who do what they do because they love math so much that they can’t imagine doing anything else. I’ve been reading a lot about math and math history in the last week, and it seems that the only thing more influential to a young mathematician than an outstanding math book is an inspiring math teacher. As it turns out, even some of the best mathematicians in the world almost gave up math entirely until they had a math teacher who emanated a love of mathematics and inspired them to greatness. A poor teacher of any subject will certainly make you want to turn your back on it, unless you have other influences to which you can turn. When you don’t have an alternative, your teacher’s attitude toward the subject greatly influences your own — and the influence is probably stronger if their attitude is negative. (He doesn’t seem happy teaching physics. So why would I want to do that and be miserable, too?)

Think of your good teachers, your inspiring teachers, who opened the door to a new world for you, and made you feel like you had every right to go through that doorway and into that new and exciting world. It didn’t have to be math, but it could have been. Maybe it was writing, or acting, or singing, or woodworking, or playing baseball. Sewing your own clothes. Separating hydrogen and oxygen in the lab. Playing the clarinet. Or watching a great teacher and deciding that being a teacher yourself was what you wanted to do.

And sometimes you have a teacher who is a delight to just have as a teacher. Their love for the subject is something special to watch even though you know you can’t and won’t follow in their footsteps. You can honor them by respecting the subject. Remember watching “Orange County Choppers”? Remember Rick the fabricator? Rick Petko didn’t get drawn into any of the family drama that fueled the ratings for the show. He just grabbed a piece of sheet metal, went to the fabricator, and did stunning work. (I had to look up his last name, but I still remembered his first name after many years because of his sheer talent.)

To this day I’m grateful when I see someone working in what is clearly their dream job. Whether they’re a social worker or a team lead or the kid taking my order at the drive-thru, you can tell they’re in exactly the right place at the right time and doing exactly what they’re supposed to do.

So right now I’m reading about math and how to think mathematically. The problems that I’m given to work on are extremely practical problems taken from real life. And as I’m trying to construct formulas that take the input and produce the output, I’m left wondering if I was ever asked to do this kind of work before. I may not have been the kid who asked, “Why are we doing this?” in math class, but I sure as heck thought it to myself. So I would like to think that if I had been asked to create a practical formula, I would remember having done it.

The reason I’m going back to work in geometry now is because I remember what it felt like when I took it in 10th grade. The geometry classes from the 1980s aren’t like the geometry classes they teach now, so there is a bit of a generation gap here. When I took the class, we studied proofs and constructions. Figuring out the logical steps and using a compass and ruler to follow directions and create a particular shape struck me as eminently practical — at the time, much more so than algebra did. In a way, I’m going back to the same textbook and the same exercises to do a little time warp and go back to when things made a bit more sense. We’ll see how that goes.

Today I said farewell to two campus friends, and I’m not sure when I’ll see them again. August? Next year? Never? Right now there’s no way to tell — and no reason to make myself upset about it. So I’m stockpiling the things that make me feel happy, grounded, and secure — until we meet again.

Flirting with math

During the last two weeks I have read a 400-page biography of the Indian mathematician Ramanujan, a 150-page memoir by (and tribute to) the British mathematician G. H. Hardy, and five chapters — just over 100 pages — of The Development of Mathematics by E. T. Bell, so it’s safe to safe that I’ve been thinking a lot about math and my relationship to it.

“Relationship” strikes me as a particularly appropriate word to use in my case. I know many people who dislike math or fear math. It’s true that some people are enchanted by math or set out to conquer math. Others battle math their whole lives and always feel that they’re on the losing side. And in the last month I have attended some academic ceremonies that included SecondBorn, so I’ve been exposed to some very young people for whom accomplishment in mathematics is just one of many notches on their belt (or feather in their cap, or flowers in their bouquet — the metaphors are endless).

My own relationship with math has been on-again, off-again, almost in the “Ross and Rachel” sense, where when one of us was in pursuit the other was running away.

In the first remembered interaction with math in the third or fourth grade, it was not that I didn’t understand what to do, but that nobody could read what I wrote. My letters and numbers were so sloppy that they were illegible, and my parents (it may have been one or the other, but I shall give the credit to both) insisted that I recopy (and recopy) my day’s homework until it met and surpassed their standard of legibility. I resented this at the time, but now I can look back at my handwriting and agree that it was the right thing for them to do. At best, my penmanship could have been termed “effusive” or “enthusiastic.” It certainly wasn’t clear, and my teachers were probably grateful for my parents’ insistence that practice would, someday, make perfect. (Mrs. DeRing? Miss Rood? What say you? And my apologies to Mrs. Anderson, who probably saw only the “before” examples.)

To illustrate how bad it was, I did a Google search for “sloppy childish homework.” I probably still have samples of my own math homework from the third and fourth grades, but this is not the night I’m going to search the house and locate the evidence. (Just because I still have everything doesn’t mean that I know exactly where it is.) Maybe later.

Not mine… but similar.

The next time I remember doing anything related to math was in the seventh grade. Mrs. Steinhauer seemed to think that I was bored, so she gave me extra work to do on magic squares. Perhaps this was just to keep me from distracting everyone else, but I enjoyed it.

Eighth grade was when my relationship with math started to get rocky. I really didn’t get this whole “negative number” thing. And there was a time in there when I got different results every time I did a math problem. I’m not sure that I can blame early teenage hormones, but I had no idea what was going on in math that year. Finally I just decided to keep doing the math problems until I got the same result twice and I’d go with that. Mr. Miller must have been incredibly tolerant of the young adolescent mind. The one thing I do remember about our middle school math books is that that they were filled with brightly colored cartoons and corny jokes to get our attention. I suppose it worked because I do remember it, but perhaps it would have been a good idea to make more of the cartoons directly related to the math we were trying to learn.

By ninth grade I was taking math for granted, reading thick novels in algebra class. I understood how to do the work (thanks, B., for teaching it to me on the bus ride home) but I didn’t find it particularly exciting or interesting.

Geometry, however, opened up a whole new world for me in tenth grade. Suddenly everything made sense. Everything meant something. And I liked doing it! Proofs? Constructions? Bring it on!

The next year I was back to algebra in the form of Algebra II, and senior year brought Advanced Math and — thanks to the gods — a semester of Computer Math class, where everything was interesting and made sense again. I converted Microsoft Basic programs to Apple Basic so they would run on the //e. I wrote my own programs. And I happily re-wrote my own programs.

Then came college, and the choice between math and English about which I’ve already written. I went with the English major in creative writing, and took a calculus class to fulfill my math requirement. I could have done this by repeating a level of math I’d already taken, but this simply didn’t occur to me. I had the sense I was supposed to go forward, and forward I went — into familiarity, then confusion. I liked calculating the lengths of the shadows of flagpoles and the number of hamsters in the nth generation, but somewhere along the line I realized that I didn’t quite understand exactly what I was doing. What was this e value, anyway? I wanted to get to know this math thing better, and really understand it. But the more I chased it, the more it slipped away from me. I went backwards in the textbook while the lecturer went forwards, and I wound up in a terrible tangle. By the time I finally went to the teacher, it was too late. She was very apologetic, but I was about ten points short of the grade she thought I deserved and there was nothing either of us could do. (Message to the college kids: visit your instructor at your first opportunity, not your last.)

Then — except for a brief flirtation with computer science, which wasn’t exactly mathematics — math and I were done. Whether we broke up or we drifted apart is hard to say.

I did remember a bit about math. At an editorial job with an engineering society, I could catch typographical errors in some high-level equations. This really impressed the authors, who had doctoral degrees in physics or engineering, but really I just knew that the equations had to be balanced. If they wanted to think that I understood thermodynamics, that was their business. But my memories of math helped me get through some tricky editorial situations.

It was more than a dozen years later when I reconnected with my ex mathematics, or perhaps I should say my x. We forgave each other quickly and got along surprisingly well as I reviewed and re-learned algebra and trigonometry. I even got a good start at another attempt at calculus right up until my personal life fell completely apart; my calculus grade was one of the sad casualties as I managed to sort out almost everything else.

In the last ten years I have flirted with data analysis, but only superficially. But I have also had the opportunity to discuss math education with a friend of mine who happens to be a mathematician. Mathematical topics are never far from his mind, and he’s always been ready to listen to my earnest thoughts — however simplistic or ill-informed — on the subject. So now I’ve been doing more reading about the history of it all, and thinking back to the kind of math I most enjoyed — geometry.

Now, I’m not considering a new career as a geometer (even if that were still a thing). But I am starting to wonder what it would be like to have some fun on the side with this old high school flame. We’ll take it one step at a time and see what we can construct.

The broader view

I’m beginning to suspect that I’ve been going about this all wrong — or, at least, all wrong for me.

All the time that I was in school, I was under the impression that I was supposed to try a lot of subjects and activities — to become well rounded, of course, but also to find out what I was particularly good at so that I could figure out, well, who I was going to be when I grew up. Decide the thing that I was going to do. Have a wide range of experiences and then… narrow things down. Focus. Aim. At… whatever my target was.

That’s harder to do when you’re interested in everything. I am fairly certain that I’ve written this before, but when you do like almost everything it makes it harder to choose what to focus on. It’s not so much deciding, “I should do this!” as realizing, “I need to give up that and that and that so I can do this.”

Apparently (she muttered as she mentally wandered through her house and glanced at her overflowing bookcases, stuffed closets, and no-space-for-a-car garage) I haven’t managed to give up much of anything at all of that and that, or that, to focus at all on this thing way over here behind the dining room table.

Instead, every time I developed a new interest I accumulated all the necessary books, tools, and materials necessary to do it to the best of my ability. And I hung on to them. The evidence surrounds me, sometimes looming over me, sometimes tripping me up. Everything I have ever wanted to do? I have a starter kit for it, right here. Just in case I ever have time to do it.

So. Here’s the thing. Right now I’m doing a lot of things that I’m happy about, but what I really don’t have is much extra time. I’m trying to squeeze in some things here and there, like learning new languages and researching an eventual move and planning a big summer trip, but day-to-day I just have time to do as much as I can and then collapse, sleep, and get up and do more the next day. So even though I want to learn the piano or practice drawing or scrapbook all my kids’ grade-school artwork, the odds are that I’m simply not going to get to it. What do I eliminate from my life so that I can concentrate on the things that “make the cut” and get to stay in my life?

That would be a great many things, and the cuts would be very painful. And since I try to keep only things that are impossible (or very difficult) to replace, eliminating several interests for the sake of a few is going to involve a series of decisions that will be very difficult (or impossible) to reverse. And what if I eliminated the wrong things?

What if that wasn’t the way I should have looked at things at all? What if I could go back in time to when I was eight years old and give myself some new guidelines? Like, you don’t have to try to be perfect at everything you try or You can read through the complete set from the library’s copies and you don’t have to buy them all or Just have fun, darn it.

I’d also say something about sticking to my core and gradually adding to it instead of trying to get rid of things to narrow my focus. I’m not sure that eight-year-old me would listen, but it would be worth trying to say. There’s so much guilt that comes with deciding to give up doing things that I wonder if it would have been better to have grown up with a different philosophy about it all.

About a week or so ago I realized that I still had a curtain — which I’m using to cover a set of shelves in our bathroom — pinned up for future sewing. It has been pinned since approximately last June because I didn’t know the exact way to finish it off perfectly. And I realized, finally, that it didn’t have to be perfect. I just needed to take it down, pull out the sewing machine, and sew the freaking seams so I could take the pins out. And if I didn’t like the way it looked I could figure out another way to do it, and I could even buy another curtain and just try again. I didn’t have to do the project in the least possible amount of time (THAT ship has sailed) to the highest possible standard (HA!) while spending the least amount of money. I just want a curtain to cover the shelves.

This might strike you as a high-quality epiphany if I didn’t go on to confess that one of those curtain panels is now folded on the edge of my dining room table and the other one is wadded on top of a box of photos I need to scan for my father’s celebration of life event at the end of July (where they have been for several months) and I have never moved my sewing machine to the table to do the work although it is a mere 27 inches from where it needs to be. NEVERTHELESS it is an epiphany, and I think that I will need to identify more areas where I have allowed the best to be the logjam of the good, when I can find the time.