On Mathematics: Multiplication Tricks and the Concept of Zero

I’m finally getting back to reading Here’s Looking at Euclid, and am determined to finish up some of the non-fiction books that have been sitting in my reading pile for an embarrassingly long time.

I’m mostly posting about chapter three because I want a place to stash this quote about Indian/Vedic multiplications tricks which I found interesting:

Vertically and Crosswise, or “cross-multiplication,” is faster, uses less space and less laborious than long multiplication. Kenneth Williams told me that whenever he explains the Vedic method to school pupils they find it easy to understand. “They can’t believe they weren’t taught it before,” he said. Schools favor long multiplication because it spells out every stage of the calculation. Vertically and Crosswise keeps some of the machinery hidden. Williams things this is no bad thing, and may even help less bright pupils. “We have to steer a path and not insist that kids have to know everything all of the time. Some kids need to know how [multiplication] works. Some don’t want to know how it works. They just want to be able to do it.” ‘

This makes sense to me because I’m a huge fan of specialization. Yes, most kids probably need to know HOW their math works, but for a kid who’s struggling with it, it probably makes more sense to teach them a trick so they can do it, and focus on honing their skills where they excel, instead of spending hours and hours on something they may never excel at, and likely teaching them to hate it in the process.

The other thing that stood out to me about this chapter was the commentary on the Indian discovery of zero being related to their religious understanding of the importance of ‘nothing’ or the void. This bothers me a little as I doubt I agree with their emphasis on nothingness, but I do think the concept of zero is important and helpful in mathematics, and would like to be able to figure out the true philosophical reason behind that.

Perhaps they simply over emphasize the importance of zero as an existent entity, and the proper understanding is closer to it actually being a placeholder for nothing, but I would love to see a Christian worldview explanation of the concept of zero, as I just can’t quite figure out where to start in exploring the philosophical underpinnings.

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Of Dreams and Plans

Every January I start to get the urge to evaluate my life and figure out what I want to be doing that I’m not doing, or at least haven’t finished achieving yet. Colton seems to mange to continuously update his plans and priorities and can’t figure out why January would have this effect on me. So this year I took a little different strategy and made a list of dreams and planned to pick one to work on every 4-6 weeks through the year.

Unsurprisingly, I’ve gotten distracted with a lot of daily life and don’t seem to make a lot of progress. According to my list the last time I picked a dream to work on was in April, and I know I didn’t actually do anything with it.

Oddly enough, a part of my dream to travel to Europe, which wasn’t even on the docket for this year, is coming true this fall. Colton has a work trip to Ireland and I get to tag along. I’m still holding out for a full tour of England, Scotland, Ireland and Wales someday, but for now, a few days in Dublin will do just fine.

Aside from travel though, I’m discovering the the core of my dreams is really to do some combination of creative writing and nutritional coaching. (Well, that and being able to sing on key.) I really enjoyed the writing I did for my food blog, and to some extent the videos I made for it, though those were also stress inducing. (Yes, random guy on the internet, it is possible to care about healthy eating and still have fat arms. FYI.) But what I really, really loved was the few comments I got about how people started using my homemade ketchup all the time, or how small children devoured green beans made from my recipe. I loved giving people one small, doable step toward living healthier.

And, honestly, that’s why I started sharing about Plexus. Because I really want to help people get healthier, and because it give me chance to talk to people about their health and nutrition one on one, and figure out personalized suggestions for what might help them.

If I do very well with Plexus maybe someday it can even fund that full Europe trip I’d love to make, but what would make me very happy is if sharing Plexus resulted in lots of stories of other people finally having energy to go do all the things they’ve always wanted to do.

Unfortunately, I don’t think Plexus will ever solve my singing off-key problems.

On Mathematics: Base Not-Ten Systems and Abacii

Chapter one (that is, the chapter following chapter zero) of Here’s Looking at Euclid started off with an extensive analysis of the base 20, base 12, base 2 (binary) and other number systems. Aside from the brief realization that when converting between TBSP and cups or cups and gallons I can actually think in an essentially base 16 system, I find it hard to wrap my head around non-base 10 systems of numbers, so on that subject I simply refer you to Toby’s defense of the base 12 system: http://geekofmanytrades.blogspot.com/2014/03/math-and-sciences-monday-cheaper-by.html

The more intriguing (or at least less brain bendingly intriguing) part of the chapter was the discussion how becoming proficient with an abacus (specifically a Japenese style soroban, with five beads to a row) can improve the speed of mental calculations. I’m beginning to wish that I learned how to properly use my childhood abacus instead of arranging the beads into patterns and rolling my eyes at manipulating the beads to show answers to basic math problems I could already do faster in my head.

After reading about the intensive mental calculations achieved by school children who master the soroban it seems like a very good way to teach basic math skills. Manipulating physical objects as a way of doing math changes which parts of the brain are used, which means that not only could it make basic math concepts much easier to grasp for more creatively and less logically minded children, but even for logically minded children, the crossover of using multiple parts of the brain should improve their ability to recall what they’ve learned. Add to this the fact that soroban then becomes a basis for much more complex calculations later, and I’m a bit tempted to go back and learn to use a soroban in my spare time myself.

Maybe right after I finally tackle quantam physics.

On Mathematics: Approximation and Exactitude

The first chapter of Here’s Looking at Euclid (which is numbered at Chapter Zero) deals with the cultural differences between dealing with numbers and instinctive human reactions to dealing with numbers, which leads to a lot of discussion of approximating numbers versus exact counting.

The author ends the chapter with the conclusion that numbers are a human construct imposed on the outside world as a way to try to make sense of it. I find this conclusion baffling as it follows on the heels of this question, “If our brains can represent numbers only approximately, then how were we able to ‘invent’ numbers in the first place?” Perhaps the whole thing would make more sense if one assumed that God created numbers and mathematics, and that even our attempts at exactness are derivative from His truly exact calculations. (Pi, anyone?)

To me, the most interesting part of this exploration of the human brain and numbers was the idea that we innately tend to think logarithmically rather than linearly. That is, we tend to think in terms of comparisons and ratios rather than exact numbers as laid out on a number line. Don’t believe me? Which sounds more drastic, the difference between one and a million or the difference between one million and two million?

See what I mean?

Even those of us who lean toward logic and precision of calculation still have a human inclination to view numbers in an approximate and comparative way. (Possibly because this is more useful in everyday life, as people who tend to get caught up on precise calculations are often reminded. Counting how many items are in the carts of each person in each line of the grocery store isn’t going to save you any time, even if you do manage to calculate which line is mathematically shortest, but a quick estimate and comparison of heaping full carts vs one nearly empty cart might save you quite a bit of time.)

Now, here’s one of the interesting bits: Teenagers who were tested on their ability to rapidly compare groups of dots and accurately estimate the differences in sizes of the groups varied greatly in their ability to make these estimates. The ones who scored highest on these tests correlated to those who tended to score highly  on their school test in the precise calculations of formal mathematics. In other words, the better you are at estimating and comparing, the better you likely are at precise calculations.

This brings to mind teaching approaches that focus on the natural developmental stages of children. Perhaps rushing children past the early, colorful, comparative stages of learning math into ‘proper’ academics actually slows down their progress in the long run. I have no idea off the top of my head what that means about teaching math as specific ages, but it does seem to lend general support in the direction of allowing younger children time to focus on creative play instead of formal academics.

This chapter of the book sparked one last ponderable thought for me: If most of our formal mathematics are based on a logical, linear scale, are there similar levels of advanced mathematics yet to be discovered along the path of more intuitive, logarithmic scale?

Always Winter

It is culturally unacceptable to be cheerful about winter time. People bundled deep in layers of coats and scarves swathe themselves with grumbling as if in added protection against the cold. They range from startled to offended if you dare to suggest that the snow is actually rather pretty. And that’s on an average winter.

When the temperatures stay low for longer than normal and the clouds are thick with any manner of winter precipitation about to drop down on our heads and power lines, the only acceptable response is grumbling. After all, clearly, if God really loved us, it would be spring. Right. Now. (And probably forever.) A few little rays of sun between the clouds are not going to cut it when we know we deserve better.

And obviously, with our clear vantage point on all eternal knowledge, we know exactly what God should do if He really loved us.

Or at least, isn’t that what we’re really saying when we complain about what God is sending us?

Now I know everyone is rolling their eyes, because I’m that crazy winter person who really does think the snow is pretty and rarely gets cold and doesn’t even like summer all that much. But before you tune me out entirely (and leaving aside the fact that even I reach the point where I start thinking green plants might just be prettier than gray slush), let me remind you that I know something about waiting.

I understand being stuck in a season that you can’t change that feels like it should have been over a long time ago. I know what it feels like to ache, not in a theoretical sense, but in a physical, ‘I wonder if this is what a heart attack feels like’ kind of pain in the chest, every time you really know that God is saying ‘not yet’. I know that feeling of convincing yourself that you’re really perfectly fine and being slightly exasperated to find tears rolling down your face anyway.

And if you think I’m going to try to downgrade Winter to the status of a lesser problem, you’re mistaken. Winter may qualify as a major trial in your life, but even if it’s a small trial I wouldn’t write it off as nothing.

Sometimes the ‘big’ trials in life are, in a way, the easiest to deal with. They come at you head on, over and over again, and you can’t ignore them or pretend that you’re not supposed to be dealing with them. I can go straight from working through big trials in my life, and finding a place of peace and calmly trusting, to being completely exasperated that I can’t find a clean pair of socks.

Because overarching life stages are one thing, and God probably has those planned out, even when I don’t like it so well. But really, I have a morning schedule to keep, and I can’t do that if I spend ten minutes trying to find a decent pair of socks to wear, and since I don’t know of any huge life altering consequences to having clean socks, clearly God just doesn’t care enough to arrange these details for me, exactly the way I want them and on my own schedule.

Clearly.

Not like there’s any obvious pride issues that need to be dealt with there or anything…

But the funny thing is, if we can just accept the fact that God really does know what He’s doing, that He really does love us, and it’s all part of this huge happy ending fairy tale He’s writing in the world, it changes how we see things. Instead of being an insult to the eternal summer we think we deserve, that tiny ray of sunlight between the clouds is a promise that it won’t be gray forever.

A Not So Perfect New Year

Every year (several days after the new year starts) I post something about how the last year went and how I’m hoping the next year will go better. This is usually accompanied by me making many lists, coming up with way too many areas in which I want to improve, and angst over the fact that it’s January and I’m already behind on my blog posts.

This year I didn’t really feel like analyzing 2013. Good stuff happened, my health improved and I failed a lot. Moving on. Rah 2014! Maybe 2014 will be the year I stop failing! More lists! More goals! More… angst…

I didn’t even plan on starting my brand new 2014 schedule on January 1st. The fact that this new clean slate of a new year begins immediately after staying up past midnight the night before has always seemed a bit ironic to me. Plus, Colton was able to get quite a few days off around Christmas and New Years this year, so I figured vacation would just stretch until he went back to work. My official new year blank slate would start on the first Monday of the new year.

On the first Saturday of the new year we discovered that we were about to have lots of snow and the coldest temperatures in a long time. We could have just holed up at home for the weekend (church had already been canceled due to the uncertainties of travel in those conditions) but the dangers of being snowed in during a record cold and then having our power go out seemed a little too risky. We could handle a power outage, and we could handle record cold, but not at the same time.

So we spent the weekend with Colton’s family, experimenting with how long we could stay out in subzero windchill throwing snowballs at each other and then proceeding to play games and watch Bleak House (which I must say, is less depressing than it always sounded from the name). Not my plan for the weekend, but fun.

This is where my plans for the new year really started to fall apart. A frozen car meant not getting back to Peoria until Monday evening, which meant skipping my new Tuesday schedule so as to do my Monday chores, notably laundry. Wednesday, the cold I’d had was finally getting quite miserable and my husband made me take a sick day. Thursday I finally caught up on dishes and got the laundry folded, but still lacking a car, grocery shopping was not going to happen. It wasn’t until today that I really got to try out my new schedule, and by then something had changed.

This wasn’t the new start to a new year I had planned, but somehow it was exactly the start I needed. It was the start of letting go of my plans and my lists and my schedules. It was the start of refusing to feel guilty for getting sick while I had work to do. It was the start that forced me to relearn what it means for God’s grace to cover my failures.

And that, I think, is an excellent way to start 2014.

Daily Thanks #18: Clean Laundry

I’m pretty sure that for my Monday daily thankfulness post (yes, I’m a little behind…) I’m supposed to write about how thankful I am to still have a house to sleep in. Or about how great it is that people in the community are stepping up to help tornado victims. Or maybe even something sweet about people getting reunited with their pets.

All those things are true, of course… But the thing is, I didn’t get scared enough during the tornado to have an epic sense of relief afterwards. My post tornado sense on Monday was along these practical lines, “Well, we can’t get out to help with disaster relief today, so I guess I’d better try to catch up on my housework as much as I can so we’re freed up to help later when they’re letting people into the areas that need help.”

The most exciting thing that happened on Monday was that for the first time in about three weeks, I got all the clean laundry folded and put away the same day it was washed.

And I guess when you come down to it, clean laundry is just exactly as mundane and as epic as it needed to be that day.