Learning from Mathematics: Society, Human Relationships, and Communication - Common Multiples and Common Divisors

Well, um, I'm not sure how to put this, but...
Hmm...
Things have been such and such.

My high school son has personally requested that I take over his studies, which until now had been entrusted to a cram school along with the tuition.

Honestly, I really wish he'd spare me.
But, you only live once. I don't want him to go off the rails at a time like this, so I've decided to take on the challenge, knowing it's a tall order.

However, my son is actually good at math. His test results are decent.
I'm looking over his past test papers to check where he's struggling or where things seem shaky.

I haven't faced things like Math I or Math A since I was a student myself.
(Though they were called something different at my school.)

I want to say, 'How should I know!', but thanks to the skills I picked up back in the day and the fact that I've been studying various things here and there, I can manage to understand it well enough at his level.

Watching him solve problems from the side, there are plenty of moments where I have to say, 'Wait a minute!'
I'm teaching him one by one, step by step, and as I show him the tricks, he's growing rapidly.

I feel like both of us have a real sense that he's making progress.
I only have one body and there are only 24 hours in a day, so even if it's a bit expensive, I'd prefer to pay someone to solve this, but I'm feeling desperate enough to think, 'Oh well! Damn it! Let's give it a try!' (laughs)

This morning, I checked his math.
It was a problem about finding the least common multiple and the greatest common divisor.

Was it a calculation error?
It was marked with an 'X'.

Try solving it again on a blank sheet of paper.
I'm watching him calculate from the side.

Maybe because he's a bit sharp, he skips the intermediate steps and does it all in his head.
He only takes notes here and there.

Ah, that's typical of someone who can't do it.

Doing things like that, he must have made a calculation error somewhere.

'Wait a minute. Write down every single step.'
He starts writing out the entire calculation process.

It might be a hassle, but this is the most reliable way.
You'll notice mistakes when you review it, and even if you make a calculation error somewhere, you can still get partial credit.

Above all, in the era these kids will live in, if you can set up the equation, you get 100 points. Calculators and computers will do the rest.

The least common multiple and the greatest common divisor.
Are you all familiar with these?

Some of you might have an allergic reaction the moment you hear the word 'mathematics' (laughs).

Please bear with me a little longer.

For example, let's say we have '6' and '8'.
The least common multiple is 24.
The greatest common divisor is 2.

The point where both numbers first meet when you multiply them is called the least common multiple.

On the other hand, the largest number that can divide both numbers evenly is the greatest common divisor.

Regarding the common divisor, for instance, if you try to divide by 3, 6 can be divided, but 8 cannot. So, it feels like 2 is the greatest.

For the least common multiple, one might be tempted to answer 6 × 8 = 48, but 24, which is in the middle, is 6 × 4 and 8 × 3. This is the closest number.

It might feel like just playing with numbers, and you might wonder if such math is ever useful, but I think it is useful in various places.

The 'building of collaborative relationships' and 'facilitation' that I usually do, especially in civic activities, contain this way of thinking.

I don't use numbers directly, but it is used as a sense.

In terms of the least common multiple,
it is like a point where multiple people with their own characteristics can join together while exerting their respective strengths, or a common goal.

In terms of the greatest common divisor,
it is like a point of agreement where people with different opinions can say, 'We can agree on that,' even while debating back and forth.

The least common multiple is something that builds up multiplicatively.
The greatest common divisor is something that shrinks divisionally.

As numbers, they are completely different directions, one being the larger and one being the smaller.

However, solving these problems involves the same method for both.
In the 'preparation stage,' you perform the same task.

That is 'prime factorization'.
It involves breaking down each element.

For example, in the case of 6 and 8,
6 = 2 × 3
8 = 2 × 2 × 2

This is how it works.

Since the only overlapping part between the two is a single 2, the greatest common divisor is 2.

On the other hand, if you want to combine them, you have to multiply the 8 by the '3' element. You also multiply the 6 by the missing '2 × 2'.
This is how the least common multiple is found.

Those who are not good at math might already have a '?' floating in their heads just looking at this article,
while those who are good at it might be thinking, 'Well, obviously' (laughs).

Based on my experience in various discussions, collaborative settings, and society, I am facing this math again for the first time in 20 years.

Even though common multiples and common divisors have opposite effects, they both require 'prime factorization' as the foundational preparation for calculation.

In other words, it is only by breaking down each element and understanding each one that common divisors and common multiples become visible.

Let's apply this to communication and society.

People who seem to have different opinions.
It feels like each of them is pointing in a different vector.

Opinions cross and cannot be reconciled.
But if you break down the opinions, you might actually find common ground.

For example, even in politics,
wanting to make this town better, wanting to make society better.
Those are parts where we can both empathize.

Even if it's just that, you can confirm that you agree on that point, and then break it down to see what you can agree on next.

The pathetic part of politics is,
there are underlying motives like 'oppose everything that is said' or 'I just don't like it'.

In terms of the least common multiple,
they have completely different abilities.
It also feels disjointed.
But if you clarify the common ground where you can empathize, and agree to at least run in the same direction for now,
a chemical reaction can occur where you decide to overlap together.
This is also the charm of 'collaboration'.

While teaching my high school son how to solve math problems,
I realized that is how to survive in society.
How to empathize with people who have different opinions.
How to work together with people who have various abilities.

I felt that he is cultivating the tools to acquire such important skills.

I myself have studied mathematics, and I feel that it has been put to use in this way.

I think that only a handful of things we study in school are actually useful in society.
However, I believe there are many things, like mental gymnastics, that can change form and be put to use.

This turned into a bit of an academic article, but please bear with me.

Well, I promised to check his physics today...
(I have to prepare for it too... (lol))
(But I'm sure I'll get angry because he hasn't studied at all... (lol))

Thank you for reading today as well.
The illustration at the beginning is a work by Masaru | note. Thank you.

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