[Subtitles: Bastiaan Cnossen]
Usually, I don't think notation in math matters that much.
Don't get me wrong,
I enjoy a bad notation rent as much as the next guy,
and there are clearly a few simple changes to our conventions
that could speed up learning for math students everywhere.
But at the end of the day, notation, good or bad, it's just not the point of math.
Even the most carefully designed symbols and syntax
will fail to capture the underlying visual that constitutes understanding.
So I figure it's better to just spend time focusing on that underlying essence
and let the symbols just be what they are in peace.
But that said, when unintuitive notation actively stalls the gears of learning
this position on the matter hardens up a little bit.
In particular, I'm thinking of one threesome of syntax, which, when you stop and think about it,
is an egregious source of friction in math education everywhere.
If you take the fact that 2 multiplied by itself 3 times equals 8 for example,
we have three separate ways to explain that relationship:
2^3 = 8, with a superscript;
∛8 = 2, with a spikely radical symbol;
and log₂(8) = 3, which we write using the word 'log' itself.
What the hell do these three ways of writing the same fact have to do with each other?
Making up syntax for a concept is fine, but don't do it in three
completely different ways for the same concept and force students to learn every
rule about that concept in three separate times.
It's like it's a different language.
This way of writing things isn't just counter-intuitive; its counter-mathematical
since rather than making seemingly different facts look the same,
which is what math should do, it takes three facts which should
obviously be the same and makes them look artificially different.
Just think about how confusing logarithms where
the first time that you learned about them.
This is of course a known issue and the Internet has no shortage of people
raising the same concern with suggestions for a better notation,
but recently I stumbled across a math exchange post with the suggestion
so lovely, so symmetrical, so utterly reasonable that I just have to share it.
For a relationship like 2 cubed equals 8, take a triangle and write 2 in the lower left,
3 on the top and 8 on the lower right.
To express the operation 2^3, remove that bottom right corner.
The symbol as a whole represents the value that should go in the missing corner.
To express log₂(8), which is asking the question
'2 to the what equals 8?', remove the top number.
The symbol as a whole represents the value that should go in the missing corner.
To express ∛8, which is saying:
'what number to the third power equals 8?',
remove the bottom left corner.
The symbol as a whole represents the value that should go in the missing corner.
In other words: all three operations are completely symmetrically represented.
This triangle deserves a name and a friend of mine at Khan Academy decided that
we should call it the triangle of power.
The definition alone is mildly pleasing but where it gets fun is when you see
how much smoother all of the different operations become.
In our current notation
there are six different ways to express the various inverse operations.
Most of these are memorized as separate entities. Some are barely even talked about.
And there's no discernible pattern, even though all of them describe the same basic idea.
But students still have to spend six times the effort to memorize each one,
are six times more likely to make a mistake,
and have six separate opportunities to decide math is dumb and boring and conducive to failure
and why don't I just go study art instead?
With the triangle of power, all these operations follow the same pattern.
Our brains are really good at picking up on patterns like this and you can much
more easily imagine a smooth mental image associated with the property.
There's even kind of an aesthetic pleasure to this and, who knows?
Maybe more of the artistically inclined students would look favorably upon that
long enough to see just how valuable their intuitions really are in the science.
Let's take another property, like the idea that a^x * a^y = a^(x+y).
The corresponding fact for logarithms is that log(xy) = log(x) + log(y)
When you write this with the triangle of power,
it's a little easier to see that both of these expressions are really saying the same thing.
Remember: the symbol as a whole represents the number at the missing corner,
so the top expression is saying that when you multiply two numbers
that belong in the bottom right of the triangle,
it corresponds with adding the numbers that belong to the top,
but that's also what the lower expression is saying:
when you multiply the numbers at the bottom right,
it corresponds with adding numbers that belong to the top.
To help students with this, you could draw inside of the triangle
saying that when the lower left is constant the numbers at the top like to add,
while the bottom right numbers like to multiply.
What about when a different corner stays constant, like the top?
Well in this case you'd write a multiplication sign in both the bottom corners,
because with exponents and radicals multiplication turns into multiplication.
The natural question that a student might ask from here
is if there's an analogous rule for when the lower right stays constant.
There is!
You have to introduce a new operation,
which for the sake of this video I'll call o-plus,
where a o-plus b equals 1 / (1/a + 1/b)
This is not actually a ridiculous thing to introduce, since it comes up in physics all the time
like when you're computing parallel resistance.
With that symbol, you could say that when the lower right number stays constant,
the top numbers like to get o-plussed together and the bottom left numbers like to get multiplied.
This is actually a really nice connection between logarithms and roots
and it never gets discussed;
probably because the notation isn't really conducive to asking the question.
I could go on and on here, showing a lot of other properties, but honestly I think
the best case I can make here is to encourage you to explore it for yourself
and notice that just about everything involving exponents, logs and radicals
becomes nicer when you use the triangle of power.
By the way, I hope that it goes without saying that in this perfect world
students wouldn't learn these operations purely from the symbols.
They should still ask why it's true and why it doesn't follow a different pattern.
But the point is that when the notation actually reflects the math,
the questions that students are most naturally asking
tend to be the ones that cut right into the essence of what's going on.
The asymmetries in the notation correspond with actual asymmetries
in the numerical relationship a^b=c itself,
not in the artificial asymmetries of squiggles and words.
When a student asks why the top likes to get added in one context, but o-plussed in another,
the teacher can point out the property that reflecting the triangle reciprocates the top
and then they can start addressing where that fact comes from.
My sincere hope is that students don't learn by symbolic patterns
but by substantive reasoning and rederivation within their own heads
but the fact is: most of us do first learn things by symbolic manipulation,
so when there's an opportunity to significantly speed up that process,
we should take it!
And if you agree with me that the triangle of power is clearly better than what we have already
start actually using it in your notes to see what it feels like.
Spread the word and if you're a teacher maybe start teaching this to your students
so that we can get them hooked while they're young.