cassandrasimplex:

basileater:

mindblowingscience:

In the remote Arctic almost 30 years ago, a group of Inuit middle school students and their teacher invented the Western Hemisphere’s first new number system in more than a century. The “Kaktovik numerals,” named after the Alaskan village where they were created, looked utterly different from decimal system numerals and functioned differently, too. But they were uniquely suited for quick, visual arithmetic using the traditional Inuit oral counting system, and they swiftly spread throughout the region. Now, with support from Silicon Valley, they will soon be available on smartphones and computers—creating a bridge for the Kaktovik numerals to cross into the digital realm.

Today’s numerical world is dominated by the Hindu-Arabic decimal system. This system, adopted by almost every society, is what many people think of as “numbers”—values expressed in a written form using the digits 0 through 9. But meaningful alternatives exist, and they are as varied as the cultures they belong to.

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this is so cool

This would have changed so much for me in grade school.

I have what was initially diagnosed as a “math-related learning disability” and would later be known as dyscalculia. It has several symptoms, like difficulty estimating distances and amounts, not understanding how long it takes to do things or how much time is passing, and other things that basically come down to “the brain keeping track of quantities”, but the primary symptom is that I can’t. fucking. do. arithmetic.

My father fought for years to keep me from saying, “I hate math.” Sure, I’d sit down at the kitchen table, look at my math homework, break into tears, and often throw up from stress. (This is “math anxiety”, which is its own separate thing anyone can develop if they experience enough trauma related to doing math, but whoo boy is dyscalculia one very certain way to develop it!) Dad insisted that I hate arithmetic, not “math”, and would only allow me to complain if I were specific about that.

We’d also play math games for fun, except Dad called them “logic games” and would try to hide the math from me until the very end. Weekend mornings, he’d crack open a Martin Gardner or Douglas Hofstadter book and pick a “puzzle” for us to work together. We’d talk it over a bit, identify important parts of the puzzle, and then he’d have me draw a picture to represent the logic I was applying.

What Dad didn’t tell me was that he, someone whose job regularly required him to do things like calculate orbital mechanics, had grown up with the same problem I was struggling with. He kept promising me that all the hard part came first, and that if I could just make it to higher forms of mathematics, I’d discover there was less and less arithmetic and more and more “logic puzzles”. He’d replace all the numbers in my math homework with letters for me, then walk me through “putting the numbers back in” one at a time when there was nothing else left to do.

So even in very early grades, if I saw a problem like

6 ÷ 3

We’d first sign a letter to the answer:

6 ÷ 3 = y

Dad would have me ask, “is 6 even or odd”? It’s even, so my problem would change to

2x ÷ 3 = y

Then he’d ask me for 6 ÷ 2 and we’d make a note off to the side that x = 3.

You might think this is where we’d put the 3 in for x in “2x”, but Dad was focused on using fewer numbers, not more. So we’d end up with

2x ÷ x = y

since if x = 3, all xs are 3s and all 3s can be replaced with xs.

“But if you’re multiplying by something, and then you divide by the same thing, all you’ve got is what you started with before you multiplied!”

That’s right, Dad would say, and he’d cross out both xs to leave us with

2 = y

“…and there’s your answer.”

This meant writing approximately three billion and twelve (but I’m bad at estimating) steps nobody else in second grade had to write down – but it also meant I made fewer mistakes and passed my math tests.

It also gave me a way to start engaged with the logic behind mathematics when the symbolic notation was working hard to “hide” the part I was good at and force me to confront, first thing, only the parts I’m neurologically unfit for.

Symbolic notation is great, don’t misunderstand me. It’s a fantastic kind of lossless compression, a shorthand in which everything has a highly specific, precise meaning. It’s SO much more difficult to write everything out in words instead. Ask me how I know.

Don’t bother; I’ll tell you anyway. I know because that’s how Dad would have me write down how I was working the math pr- uh, logic puzzles we did together. And a decade plus later, when I was volunteering as a math tutor at community college to work with other students who had math learning disabilities, I’d ask them to do the same thing.

Sure, it takes six hundred and fifty-two times as long (curse you, estimation!) as just doing the problem once. But it actually doesn’t take longer than doing the problem WRONG twelve times, getting twelve different answers, not knowing whether the right answer is among them at all (let alone which one it might be), getting frustrated, taking “a break” and getting up to do something else, coming back to it e next morning and trying again only to get a thirteenth answer of unknown quality, throwing the whole paper into the trash (and considering setting it on fire), and then nothing else because you can’t bring yourself to prioritize that task again until class time, when you try to “just quickly do it to have something to turn in” and making a zero for something you handed in instead of something you didn’t, further permanently centering your belief that you’re “just bad at math”.

So eventually, I learned I could go the long route, get an answer I could trust, actually learn whatever skill the homework was trying to teach, and get credit for my work… or I could spin my wheels in the mud for roughly equal hours producing nothing, learning nothing, and failing the class by way of failing every single assignment I was given. While crying and throwing up.

And the other students I tutored in college learned the same thing. (They all passed. They ALL passed. You have NO IDEA how proud I am of that, for myself and for them!)

Turns out, “math” is just a way of taking about logic in shorthand. But it’s the logic that counts most, not the shorthand! And when there’s a complete disconnect between the symbolic notation and the logic it represents, some people who have no trouble at all with mathematical logic are going to fail because their brains get lost in the symbols. When your brain gets left behind by the symbols, you can’t learn the logic.

I have yet to meet anyone who’s just innately “bad at math”. Y'all. I need you to read that again. Three times. Four times, if your reflexive response is “Bullshit, I’M bad at math.” I have yet to meet anyone who is innately bad at math, and I say that as someone who is neurologically incapable of adding one and one reliably. I’ve met people with math anxiety (SO MANY people with math anxiety) and people with low frustration tolerance and lots of other people with dyscalculia. I’ve met people bad at memorizing and people with less-common learning styles whose brains can’t handle learning something intimidating at the same time they’re struggling to translate from, say, spoken information to a visual or tactile-kinesthetic leaning style.

But “bad at math”? Nope. Never yet. Not even me.

So why am I going off about dyscalculia on a post about Inuit numerals?

Because this is symbolic notation with the logic baked right in.

This is symbolic notation that TEACHES the logic of arithmetic rather than serving to obscure it.

This is notation that represents the logic clearly to both visual and tactile-kinesthetic learners.

Look at that long division. LOOK AT IT.

Actually, look at the subtraction and addition examples first. Then look at the long division. If the first two examples make great sense, but the long division makes your head swim, I’m going to ask you to consider something: that you don’t really understand the logic behind long division as well as you think you do. Not as well as you understand the logic behind addition and subtraction.

I mean, you understand it will enough to perform it. Probably. Although lbr, when was the last time you had to write out a long division problem and work it by hand? You probably learned to manipulate the symbols, passed that grade, and have used a calculator for anything remotely like long division ever since the school system decided you no longer needed “practice” at it.

But you might not have learned it DEEPLY. You might not have learned all it has to say about the way we count, about the way the decimal system works and about how other systems of counting could work instead. Which is fine, in a way, for adults; we get very entrenched in our hatred for math and our resentment that it can benefit us to learn more about it, to truly understand it. We have other tools we rely on and maybe the idea of going back and re-learning how to do long division makes our teeth grind. Because long division sucks, right?

Most of us hate long division, if we’re honest.

Unless we had fathers, teachers, tutors, mentors, or friends who wouldn’t take “I hate math” for an answer and demanded we separate hatred of calculation from hatred of logic. Or just being bad at calculation from being “bad at math”.

But the thing I learned as a tutor is that everyone who hates math, everyone who believes they’re bad at math, they got left behind at some point. They missed one day of class when the logic of something like long division was being taught, and when they got back, they memorized how to go through the steps, but they stopped really grokking the logic in fullness. And after that, anything else that built on that skill that got left behind will also be something they have to just memorize, not understand.

More than half my job as a tutor was “rewinding” to whatever early skill I could find that the student couldn’t explain and re-teaching that – and then just getting the hell out of their way as everything else they’d memorized that built on that fell into place as true understanding. So fast. SO FAST. It’s like watching a video of a jigsaw puzzle assembling itself at triple speed.

I had to pause up there at “more than half my job” to reconsider the phrase, because more than half my job was also rebuilding lost confidence to battle math anxiety, but you know… in reality, fixing those left-behind points was a huge part of that confidence-building, because that’s the point when the student fully realizes that HOW THEY WERE TAUGHT had more to do with their poor performance in math classes than anything else, including dyscalculia. So no, it stands – even if fixing the left-behind points was less than half the job directly, it’s also a critical step in doing the other major task of restoring confidence.

And ever since then, I get SO angry about math education sometimes, because I know from my own experiences that even a “hey presto, you’re permanently Bad At Arithmetic Forever!” learning durability that can’t be changed isn’t enough to make anyone actually, permanently and forever bad-at-math. And I know from my father’s experience that even that kind of leaning disability isn’t, all on its own, enough to keep someone from being very successful in an EXTREMELY math-centric career for fifty years. And loving it! Truly enjoying it! And I know from the experiences of students I’ve tutored, as well as my own, that the logic of math is one long, unbroken chain (well, actually it branches into several later on) where any one link being weak, especially an early one, can completely wreck a person’s ability to enjoy or succeed at math for multiple decades afterward, and that it’s very easy for one link of that logic to just kind of fail to form in a way that goes unnoticed, again for multiple decades.

Some of the people I tutored were retirees. Some of them were great-grandparents. You get lots of older people returning to community colleges after decades of believing themselves “bad at math”. I saw how angry THEY got when they realized that something like learning how to cancel fractions without understanding WHY to cancel fractions had been not just holding them back in all math that followed but wrecking their self-image and giving them panic attacks for longer than I, at that point, had been alive. Their rage became mine.

So when I say that this is important. When I say that this visual representation of arithmetic is important.

I wouldn’t want to try to do calculus in this system. It’s not going to mesh well with established notation. I definitely wouldn’t want to use it in any algebra that uses x, y, m, n, w, and v as common variables. I wouldn’t want to try to use it in exponential notation. Writing something like “10^5” would be a nightmare to keep legible with numerals based in part on “^” as a number instead of an operand! But for teaching arithmetic itself? This isn’t just alternate notation for numbers, it’s a METHOD. Just like the sign you use for long division. Just like using sigma for sums. It’s numeric representation that also encapsulates logic like an operand does. Squeeze this in between “counting on the number line” (already a visual representation of logic that students discard as soon as they’ve mastered it) and solving expressions like “7 + 3” and fewer students get left behind so early they never recover from it.

So like. It’s important. It’s important for reasons already stated, like its cultural significance. It’s important because it contains a lesson about how cultures oppressed nearly to extinction matter and have ideas of value, ones they came up with long ago, that are still useful but end up ignored thanks to racism and xenophobia.

But like also. Get this the fuck into classrooms, I beg us all. THIS IS A GIFT. Billions, literally billions, of people could benefit from this. It isn’t just a cultural curiosity – and I feel like there’s still some lingering patronizing going on here: aw, isn’t it cute what these Inuit kids came up with, see how smart other cultures can STOP THAT. This is vital, it is modern, IT. IS. NEEDED. This is research into the advancement of mathematics as a field that must be taught before it can be practiced.

(Source: scientificamerican.com, via cassandrasimplex)

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